MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS., where

Size: px
Start display at page:

Download "MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS., where"

Transcription

1 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN -UNIFORM HYPERGRAPHS ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Abstract. We investigate minimum vertex degree conditions for -uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every -uniform n-vertex (n even hypergraph H with minimum vertex degree δ (H ( o( ( n contains a loose Hamilton cycle. This bound is asymptotically best possible.. Introduction We consider k-uniform hypergraphs H, that are pairs H = (V, E with vertex sets V = V (H and edge sets E = E(H ( ( V k, where V k denotes the family of all k-element subsets of the set V. We often identify a hypergraph H with its edge set, i.e., H ( V k, and for an edge {v,..., v k } H we often suppress the enclosing braces and write v... v k H instead. Given a k-uniform hypergraph H = (V, E and a set S = {v,..., v s } ( V s let deg(s = deg(v,..., v s denote the number of edges of H containing the set S and let N(S = N(v,..., v s denote the set of those (k s-element sets T ( V k s such that S T forms an edge in H. We denote by δ s (H the minimum s-degree of H, i.e., the minimum of deg(s over all s-element sets S V. For s = the corresponding minimum degree δ (H is referred to as minimum vertex degree whereas for s = k we call the corresponding minimum degree δ k (H the minimum collective degree of H. We study sufficient minimum degree conditions which enforce the existence of spanning, so-called Hamilton cycles. A k-uniform hypergraph C is called an l- cycle if there is a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, every vertex is contained in an edge and two consecutive edges (where the ordering of the edges is inherited by the ordering of the vertices intersect in exactly l vertices. For l = we call the cycle loose whereas the cycle is called tight if l = k. Naturally, we say that a k-uniform, n-vertex hypergraph H contains a Hamilton l-cycle if there is a subhypergraph of H which forms an l-cycle and which covers all vertices of H. Note that a Hamilton l-cycle contains exactly n/(k l edges, implying that the number of vertices of H must be divisible by (k l which we indicate by n (k ln. The second author was supported by GIF grant no. I /005. The third author was supported through the Heisenberg-Programme of the Deutsche Forschungsgemeinschaft (DFG Grant SCHA 6/4-. The collaboration of the authors was supported by joint grant of the Deutschen Akademischen Austausch Dienst (DAAD and the Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível (CAPES.

2 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Minimum collective degree conditions which ensure the existence of tight Hamilton cycles were first studied in [6] and in [, ]. In particular, in [, ] Rödl, Ruciński, and Szemerédi found asymptotically sharp bounds for this problem. Theorem. For every k and γ > 0 there exists an n 0 such that every k- uniform hypergraph H = (V, E on V = n n 0 vertices with δ k (H (/+γn contains a tight Hamilton cycle. The corresponding question for loose cycles was first studied by Kühn and Osthus. In [0] they proved an asymptotically sharp bound on the minimum collective degree which ensures the existence of loose Hamilton cycles in -uniform hypergraphs. This result was generalised to higher uniformity by the last two authors [4] and independently by Keevash, Kühn, Osthus and Mycroft in [7]. Theorem. For all integers k and every γ > 0 there exists an n 0 such that every k-uniform hypergraph H = (V, E on V = n n 0 vertices with n (k N and δ k (H ( (k + γn contains a loose Hamilton cycle. Indeed, in [4] asymptotically sharp bounds for Hamilton l-cycles for all l < k/ were obtained. Later this result was generalised to all 0 < l < k by Kühn, Mycroft, and Osthus [9]. These results are asymptotically best possible for all k and 0 < l < k. Hence, asymptotically, the problem of finding Hamilton l-cycles in uniform hypergraphs with large minimum collective degree is solved. The focus of this paper are conditions on the minimum vertex degree which ensure the existence of Hamilton cycles. For δ (H very few results on spanning subhypergraphs are known (see e.g. [, ]. In this paper we give an asymptotically sharp bound on the minimum vertex degree in -uniform hypergraphs which enforces the existence of loose Hamilton cycles. Theorem (Main result. For all γ > 0 there exists an n 0 such that the following holds. Suppose H is a -uniform hypergraph on n > n 0 with n N and ( ( 7 n δ (H > 6 + γ. Then H contains a loose Hamilton cycle. In the proof we apply the so-called absorbing technique. In [] Rödl, Ruciński, and Szemerédi applied this elegant approach to tackle minimum degree problems for spanning graphs and hypergraphs. In our case it reduces the problem of finding a loose Hamilton cycle to the problem of finding a nearly spanning loose path and indeed, finding such a path will be the main obstacle to Theorem. As mentioned above, Theorem is best possible up to the error constant γ as seen by the following construction from [0]. Fact 4. For every n N there exists a -uniform hypergraph H = (V, E on V = n vertices with δ (H 7 6( n O(n, which does not contain a loose Hamilton cycle. Proof. Consider the following -uniform hypergraph H = (V, E. Let A B = V be a partition of V with A = n 4 and let E be the set of all triplets from V with at least one vertex in A. Clearly, δ (H = ( A + A ( B = 7 n 6( O(n. Now consider an arbitrary cycle in H. Note that every vertex, in particular every

3 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES vertex from A, is contained in at most two edges of this cycle. Moreover, every edge of the cycle must intersect A. Consequently, the cycle contains at most A < n/ edges and, hence, cannot be a Hamilton cycle. We note that the construction H in Fact 4 satisfies δ (H n/4 and indeed, the same construction proves that the minimum collective degree condition given in Theorem is asymptotically best possible for the case k =. This leads to the following conjecture for minimum vertex degree conditions enforcing loose Hamilton cycles in k-uniform hypergraphs. Let k and let H k = (V, E be the k-uniform, n-vertex hypergraph on V = A B with A = n (k. Let E consist of all k-sets intersecting A in at least one vertex. Then H k does not contain a loose Hamilton cycle and we believe that any k-uniform, n-vertex hypergraph H which has minimum vertex degree δ (H δ (H k + o(n contains a loose Hamilton cycle. Indeed, Theorem verifies this for the case k =.. Proof of the main result The proof of Theorem will be given in Section.. It uses several auxiliary lemmas which we introduce in Section.. We start with an outline of the proof... Outline of the proof. We will build a loose Hamilton cycle by connecting loose paths. Formally, a -uniform hypergraph P is a loose path if there is an ordering (v,..., v t of its vertices such that every edge consists of three consecutive vertices, every vertex is contained in an edge and two consecutive edges intersect in exactly one vertex. The elements v and v t are called the ends of P. The Absorbing Lemma (Lemma 7 asserts that every -uniform hypergraph H = (V, E with sufficiently large minimum vertex degree contains a so-called absorbing loose path P, which has the following property: For every set U V \ V (P with U N and U βn (for some appropriate 0 < β < γ there exists a loose path Q with the same ends as P, which covers precisely the vertices V (P U. The Absorbing Lemma reduces the problem of finding a loose Hamilton cycle to the simpler problem of finding an almost spanning loose cycle, which contains the absorbing path P and covers at least ( βn of the vertices. We approach this simpler problem as follows. Let H be the induced subhypergraph H, which we obtain after removing the vertices of the absorbing path P guaranteed by the Absorbing Lemma. We remove from H a small set R of vertices, called the reservoir (see Lemma 6, which has the property that many loose paths can be connected to one loose cycle by using the vertices of R only. Let H be the remaining hypergraph after removing the vertices from R. We will choose P and R small enough, so that δ (H ( o( ( V (H. The third auxiliary lemma, the Path-tiling Lemma (Lemma 0, asserts that all but o(n vertices of H can be covered by a family of pairwise disjoint loose paths and, moreover, the number of those paths will be constant (independent of n. Consequently, we can connect those paths and P to form a loose cycle by using exclusively vertices from R. This way we obtain a loose cycle in H, which covers all but the o(n left-over vertices from H and some left-over vertices from R. We will ensure that the number of those yet uncovered vertices will be smaller than βn and, hence, we can appeal to the absorption property of P and obtain a Hamilton cycle.

4 4 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT As indicated earlier, among the auxiliary lemmas mentioned above the Pathtiling Lemma is the only one for which the full strength of the condition ( o( ( n is required and indeed, we consider Lemma 0 to be the main obstacle to proving Theorem. For the other lemmas we do not attempt to optimise the constants... Auxiliary lemmas. In this section we introduce the technical lemmas needed for the proof of the main theorem. We start with the connecting lemma which is used to connect several short loose paths to a long one. Let H be a -uniform hypergraph and (a i, b i i [k] a set consisting of k mutually disjoint pairs of vertices. We say that a set of triples (x i, y i, z i i [k] connects (a i, b i i [k] if i [k] {a i, b i, x i, y i, z i } = 5k, i.e. the pairs and triples are all disjoint, for all i [k] we have {a i, x i, y i }, {y i, z i, b i } H. Suppose that a and b are ends of two disjoint loose paths not intersecting {x, y, z} and suppose that (x, y, z connects (a, b. Then this connection would join the two paths to one loose path. The following lemma states that several paths can be connected, provided the minimum vertex degree is sufficiently large. Lemma 5 (Connecting lemma. Let γ > 0, let m be an integer, and let H = (V, E be a -uniform hypergraph on n vertices with δ (H ( 4 + γ ( n and n γm/. For every set (a i, b i i [m] of mutually disjoint pairs of distinct vertices, there exists a set of triples (x i, y i, z i i [m] connecting (a i, b i i [m]. Proof. We will find the triples (x i, y i, z i to connect a i with b i for i [m] inductively as follows. Suppose, for some j < k the triples (x i, y i, z i with i < j are constructed so far and for (a, b = (a j, b j we want to find a triple (x, y, z to connect a and b. Let ( m U = V \ i= j {a i, b i } i= {x i, y i, z i } and for a vertex u V let L u = (V \ {u}, E u be the link graph of v defined by E u = {vw : uvw E(H}. We consider L a [U] and L b [U], the subgraphs of L a and L b induced on U. Owing to the minimum degree condition of H and to the assumption m γn/, we have ( ( ( n e(l a [U] 4 + γ 5m(n 4 + γ ( n ( 6 and the same lower bound also holds for e(l b [U]. Note that any pair of edges xy L a [U] and yz L b [U] with x z leads to a connecting triple (x, y, z for (a, b. Thus, if no connecting triple exists, then for every vertex u U one of the following must hold: either u is isolated in L a [U] or L b [U] or it is adjacent to exactly one vertex w in both graphs L a [U] and L b [U]. In other words, any vertex not isolated in L a [U] has at most one neighbour in L b [U]. Let I a be the set of isolated vertices

5 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 5 in L a [U]. Since e(l a [U] > 4( n we have Ia < n/. Consequently, ( Ia e(l b [U] + {uw E(L b [U]: u U \ I a } ( ( Ia n/ + ( U I a < + n. Using γ /4 and n γ/ we see that this upper bound violates the lower bound on e(l b [U] from (. When connecting several paths to a long one we want to make sure that the vertices used for the connection all come from a small set, called reservoir, which is disjoint to the paths. The existence of such a set is guaranteed by the following. Lemma 6 (Reservoir lemma. For all 0 < γ < /4 there exists an n 0 such that for every -uniform hypergraph H = (V, E on n > n 0 vertices with minimum vertex degree δ (H ( 4 + γ ( n there is a set R of size at most γn with the following property: For every system (a i, b i i [k] consisting of k γ n/ mutually disjoint pairs of vertices from V there is a triple system connecting (a i, b i i [k] which, moreover, contains vertices from R only. Proof. We shall show that a random set R has the required properties with positive probability. For this proof we use some exponential tail estimates. Here we will follow a basic technique described in [5, Section.6]. Alternatively Lemma 6 could be deduced more directly from Janson s inequality. For given 0 < γ < /4 let n 0 be sufficiently large. Let H be as stated in the lemma and v V (H. Let L(v be the link graph defined on the vertex set V (H \ {v}, having the edges e E(L(v if e {v} H. Note that L(v contains deg H (v edges. Since the edge set of the omplete graph K n can be decomposed into n edge disjoint matchings, we can decompose the edge set of L into i 0 = i 0 (v < n pairwise edge disjoint matchings. We denote these matchings by M (v,..., M i0 (v. We randomly choose a vertex set V p from V by including each vertex u V into V p with probability p = γ γ independently. For every i [i 0 ] let X i (v = M i (v denote the number of edges e M i (v contained in V p. This way X i (v is a binomially distributed random variable with parameters M i (v and p. Using the following Chernoff bounds for t > 0 (see, e.g., [5, Theorem.] ( Vp we see that P [Bin(m, ζ mζ + t] < e t /(ζm+t/ ( P [Bin(m, ζ mζ t] < e t /(ζm ( γ n V p pn + (n ln 0 / γn k (4 with probability at least 9/0. Further, using ( and M i (v n/ we see that with probability at most n there exists an index i [i 0 ] such that X i (v M i (v p (n ln n /. Using i [i 0] M i(v = deg H (v and recalling that deg Vp (v denotes the degree of v in

6 6 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT H[V p {v}] we obtain that deg Vp (v = i [i 0] X i (v p deg H (v n(n ln n /, (5 holds with probability at least n. Repeating the same argument for every vertex v V we infer from the union bound that (5 holds for all vertices v V simultaneously with probability at least /n. Hence, with positive probability we obtain a set R satisfying (4 and (5 for all v V. Let (a i, b i i [k] be given and let S = i [k] {a i, b i }. Then we have R S γn and ( ( ( ( γn R S deg R S (v deg R (v 4 + γ 4 + γ for all v V. Thus, we can appeal to the Connecting Lemma (Lemma 5 to obtain a triple system which connects (a i, b i i [k] and which consists of vertices from R only. Next, we introduce the Absorbing Lemma which asserts the existence of a short but powerful loose path P which can absorb any small set U V \ V (P. In the following note that ( 5 8 < 7 6. Lemma 7 (Absorbing lemma. For all γ > 0 there exist β > 0 and n 0 such that the following holds. Let H = (V, E be a -uniform hypergraph on n > n 0 vertices which satisfies δ (H ( γ ( n. Then there is a loose path P with V (P γ 7 n such that for all subsets U V \ V (P of size at most βn and U N there exists a loose path Q H with V (Q = V (P U and P and Q have exactly the same ends. The principle used in the proof of Lemma 7 goes back to Rödl, Ruciński, and Szemerédi. They introduced the concept of absorption, which, roughly speaking, stands for a local extension of a given structure, which preserves the global structure. In our context of loose cycle we say that a 7-tuple (v,..., v 7 absorbs the two vertices x, y V if v v v, v v 4 v 5, v 5 v 6 v 7 H and v xv 4, v 4 yv 6 H are guaranteed. In particular, (v,..., v 7 and (v, v, v, x, v 4, y, v 6, v 5, v 7 both form loose paths which, moreover, have the same ends. The proof of Lemma 7 relies on the following result which states that for each pair of vertices there are many 7-tuples absorbing this pair, provided the minimum vertex degree of H is sufficiently large. Proposition 8. For all 0 < γ < /8 there exists an n 0 such that the following holds. Suppose H = (V, E is a -uniform hypergraph on n > n 0 vertices which satisfies δ (H ( γ ( n, then for every pair of vertices x, y V the number of 7-tuples absorbing x and y is at least (γn 7 /8. Proof. For given γ > 0 we choose n 0 = 68/γ 7. First we show the following. Claim 9. For every pair x, y V (H of vertices there exists a set D = D(x, y V of size D = γn such that one of the following holds: deg(x, d γn and deg(y, d 8n for all d D or

7 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 7 deg(y, d γn and deg(x, d 8n for all d D. Proof of Claim 9. By assuming the contrary there exists a pair x, y such that no set D = D(x, y fulfills Claim 9. Let A(z = {d V : deg(z, d < γn} and let a = A(x /n and b = A(y /n. Without loss of generality we assume that a b. Note that there are at most (a + γn vertices v V satisfying deg(y, v 8n. Let B(y = {v V : deg(y, v n/8} and note that B(y < (a + γn. Hence, the number of ordered pairs (u, v such that u B(y and {u, v, y} H is at most B(y (n A(y + A(y γn (a + γ( bn + bγn. Consequently, with deg(y being the number of ordered pairs (u, v such that {u, v, y} H we have ( (5 + 9γ n deg(y ( b a γ n + (a + γ( bn + bγn Hence, we obtain ( (5 + 9γ 8 8 n 8 ( + 8γn (5a b 8ab +, 8 ( n deg(y n 5 ( b + (a + γ 8 8 b n + bγn n ( + 8γn (5a b 8ab +, 8 8 where in the last inequality we use the fact that b /8 which is a direct consequence of the condition on δ (H. It is easily seen that this maximum is attained by a = b = /8, for which we would obtain ( (5 deg(y + γ n, 8 a contradiction. We continue the proof of Proposition 8. For a given pair x, y V we will select the tuple v,..., v 7 such that the edges v v v, v v 4 v 5, v 5 v 6 v 7 H and v xv 4, v 4 yv 6 H are guaranteed. Note that (v,..., v 7 forms a loose path with the ends v and v 7 and (v, v, v, x, v 4, y, v 6, v 5, v 7 also forms a loose path with the same ends, showing that (v,..., v 7 is indeed an absorbing tuple for the pair a, b. Moreover, we will show that the number of choices for each v i will give rise to the number of absorbing tuples stated in the proposition. First, we want to choose v 4 and let D(x, y be a set with the properties stated in Claim 9. Without loss of generality we may assume that N(y, d 8n for all d D(x, y. Fixing some v 4 D(x, y We choose v N(x, v 4 for which there are γn choices. This gives rise to to hyperegde v xv 4 H. and applying Claim 9 to v and v 4 we obtain a set D(v, v 4 with the properties stated in Claim 9 and we choose v D(v, v 4. We choose v N(v, v to obtain the edge v v v H. Note that N(v, v γn. Next, we choose v 5 from N(v, v 4 which has size N(v, v 4 γn. This gives rise to the edge v v 4 v 5 H. We choose v 6 from

8 8 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT N(y, v 4 with the additional property that deg(v 5, v 6 γn/. Hence, we obtain v 4 yv 6 H and we claim that there are at least γn/ such choices. Otherwise at least ( N(y, v 4 γn/ vertices v V satisfy deg(v 5, v < γn/, hence deg(v 5 < γ 6 n + ( ( γ n < δ(h, which is a contradiction. Lastly we choose v 7 N(v 5, v 6 to obtain the edge v 5 v 6 v 7 H which completes the absorbing tuple (v,..., v 7. The number of choices for v,..., v 7 is at least (γn 7 /4 and there are at most n 6 choices such that v i = v j for some i j. Hence, we obtain at least (γn 7 /8 ( 7 absorbing 7-tuples for the pair x, y. With Proposition 8 and the connecting lemma (Lemma 5 at hand the proof of the absorbing lemma follows a scheme which can be found in [, 4]. We choose a family F of 7-tuples by selecting each 7-tuples with probability p = γ 7 n 6 /448 independently. Then, it is easily shown that with non-zero probability the family F satisfies F γ 7 n/, for all pairs x, y V there are at least pγ 7 n 7 /6 tuples in F which absorbs x, y the number of intersecting pairs of 7-tuples in F is at most pγ 7 n 7 / We eliminate intersecting pairs of 7-tuples by deleting one tuple for each such pair. By definition each for the remaining 7-tuples (v, i..., v7 i i [k] with k γ 7 n/ forms a loose path with ends v i and v7 i and appealing to Lemma 5 we can connect them to one loose path which can absorb any pγ 7 n 7 / = β pairs of vertices, proving the lemma. To avoid unnecessary calculations we omit the details here. The next lemma is the main obstacle when proving Theorem. It asserts that the ( vertex set of a -uniform hypergraph H with minimum vertex degree δ (H o( ( n can be almost perfectly covered by a constant number of vertex disjoint loose paths. Lemma 0 (Path-tiling lemma. For all γ > 0 and α > 0 there exist integers p and n 0 such that for n > n 0 the following holds. Suppose H is a -uniform hypergraph on n vertices with minimum vertex degree δ (H ( γ ( n. Then there is a family of p disjoint loose paths in H which covers all but at most αn vertices of H. The proof of Lemma 0 uses the weak regularity lemma for hypergraphs and will be given in Section... Proof of the main theorem. In this section we give the proof of the main result, Theorem. The proof is based on the three auxiliary lemmas introduced in Section. and follows the outline given in Section.. Proof of Theorem. For given γ > 0 we apply the Absorbing Lemma (Lemma 7 with γ/8 to obtain β > 0 and n 7. Next we apply the Reservoir Lemma (Lemma 6 for γ = min{β/, γ/8} to obtain n 6 which is n 0 of Lemma 6. Finally, we apply the

9 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 9 Path-tiling Lemma (Lemma 0 with γ/ and α = β/ to obtain p and n 0. For n 0 of the theorem we choose n 0 = max{n 7, n 6, n 0, 4(p + (γ }. Now let n n 0, n N and let H = (V, E be a -uniform hypergraph on n vertices with ( ( 7 n δ (H 6 + γ. Let P 0 H be the absorbing path guaranteed by Lemma 7. Let a 0 and b 0 be the ends of P 0 and note that V (P 0 γ n < γn/8. Moreover, the path P 0 has the absorption property, i.e., for all U V \ V (P 0 with U βn and U N there exists a loose path Q H s.t. V (Q = V (P 0 U and Q has the ends a 0 and b 0. (6 Let V = (V \ V (P 0 {a 0, b 0 } and let H = H[V ] = (V, E(H ( V be the induced subhypergraph of H on V. Note that δ (H ( γ( n. Due to Lemma 6 we can choose a set R V of size at most γ V γ n such that for every system consisting of at most (γ V / mutually disjoint pairs of vertices from V can be connected using vertices from R only. Set V = V \ (V (P 0 R and let H = H[V ] be the induced subhypergraph of H on V. Clearly, ( 7 δ(h 6 + γ ( n Consequently, Lemma 0 applied to H (with γ 0 and α yields a loose path tiling of H which covers all but at most α V αn vertices from V and which consists of at most p paths. We denote the set of the uncovered vertices in V by T. Further, let P, P..., P q with q p denote the paths of the tiling. By applying the reservoir lemma appropriately we connect the loose paths P 0, P,..., P q to one loose cycle C H. Let U = V \ V (C be the set of vertices not covered by the cycle C. Since U R T we have U (α + γ 6 n βn. Moreover, since C is a loose cycle and n N we have U N. Thus, using the absorption property of P 0 (see (6 we can replace the subpath P 0 in C by a path Q (since P 0 and Q have the same ends and since V (Q = V (P 0 U the resulting cycle is a loose Hamilton cycle of H.. Proof of the Path-tiling Lemma In this section we give the proof of the Path-tiling Lemma, Lemma 0. Lemma 0 will be derived from the following lemma. Let M be the -uniform hypergraph defined on the vertex set {,..., 8} with the edges, 45, 456, 678 M. We will show that the condition δ (H ( o( ( n will ensure an almost perfect M-tiling of H, i.e., a family of vertex disjoint copies of M, which covers almost all vertices. Lemma. For all γ > 0 and α > 0 there exists n 0 such that the following holds. Suppose H is a -uniform hypergraph on n > n 0 vertices with minimum vertex degree ( ( 7 n δ (H 6 + γ. Then there is an M-tiling of H which covers all but at most αn vertices of H.

10 0 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT The proof of Lemma requires the regularity lemma which we introduce in Section.. Sections. and. are devoted to the proof of Lemma and finally, in Section.4, we deduce Lemma 0 from Lemma by making use of the regularity lemma... The weak regularity lemma and the cluster hypergraph. In this section we introduce the weak hypergraph regularity lemma, a straightforward extension of Szemerédi s regularity lemma for graphs [5]. Since we only apply the lemma to -uniform hypergraphs we will restrict the introduction to this case. Let H = (V, E be a -uniform hypergraph and let A, A, A be mutually disjoint non-empty subsets of V. We define e(a, A, A to be the number of edges with one vertex in each A i, i [], and the density of H with respect to (A, A, A as d(a, A, A = e H(A, A, A. A A A We say the triple (V, V, V of mutually disjoint subsets V, V, V V is (ε, d- regular, for constants ε > 0 and d 0, if d(a, A, A d ε for all triple of subsets A i V i, i [], satisfying A i ε V i. The triple (V, V, V is called ε-regular if it is (ε, d-regular for some d 0. It is immediate from the definition that an (ε, d-regular triple (V, V, V is (ε, d-regular for all ε > ε and if V i V i has size V i c V i, then (V, V, V is (ε/c, d-regular. Next we show that regular triples can be almost perfectly covered by copies of M provided the sizes of the partition classes obey certain restrictions. First note that M is a subhypergraph of a tight path. The latter is defined similarly to a loose path, i.e. there is an ordering (v,..., v t of the vertices such that every edge consists of three consecutive vertices, every vertex is contained in an edge and two consecutive edges intersect in exactly two vertices. Proposition. Suppose H is a -uniform hypergraph on m vertices with at least dm edges. Then there is a tight path in H which covers at least (dm + vertices. In particular, if H is -partite with the partition classes V, V, V and dm > 0 then for each i [] there is a copy of M in H which intersects V i in exactly two vertices and the other partition classes in three vertices. Proof. Starting from H we remove all edges containing u, v for each pair u, v V of vertices such that 0 < deg(u, v < dm. We keep doing this until every pair u, v satisfies deg(u, v = 0 or deg(u, v dm in the current hypergraph H. Since less than ( m dm < dm e(h edges were removed during the process we know that H is not empty. Hence we can pick a maximal non-empty tight path (v, v,..., v t in H. Since the pair v, v is contained in an edge in H it is contained in dm edges and since the path was chosen to be maximal all these vertices must lie in the path. Hence, the chosen tight path contains at least (dm + vertices. This completes the first part of the proof. For the second part, note that there is only one way to embed a tight path into a -partite -uniform hypergraph once the two starting vertices are fixed. Since M is a subhypergraph of the tight path on eight vertices we obtain the second part of the statement by possibly deleting up to two starting vertices.

11 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES Proposition. Suppose the triple (V, V, V is (ε, d-regular with d ε and suppose the sizes of the partition classes satisfy m = V V V with 5 V ( V + V (7 and ε m > 7. Then there is an M-tiling of (V, V, V leaving at most εm vertices uncovered. Proof. Note that if we take a copy of M intersecting V i, i [] in exactly two vertices then this copy intersects the other partition classes in exactly three vertices. We define t i = ( ε 8 ( V j + V k 5 V i where i, j, k [] are distinct. Due to our assumption all t i are non-negative and we choose t i copies of M intersecting V i in exactly two vertices. This would leave V i (t i + t j + t k = ε V i vertices in V i uncovered, hence at most εm in total. To complete the proof we exhibit a copy of M in all three possible types in the remaining hypergraph, hence showing that the choices of the copies above are indeed possible. To this end, from the remaining vertices of each partition class V i take a subset U i, i [] of size ε V i. Due to the regularity of the triple (V, V, V we have e(u, U, U (d ε(εm. Hence, by Proposition there is a copy of M (of each type in (U, U, U. The connection of regular partitions and dense hypergraphs is established by regularity lemmas. The version introduced here is a straightforward generalisation of the original regularity lemma to hypergraphs (see, e.g., [,, 4]. Theorem 4. For all t 0 0 and ε > 0, there exist T 0 = T 0 (t 0, ε and n 0 = n 0 (t 0, ε so that for every -uniform hypergraph H = (V, E on n n 0 vertices, there exists a partition V = V 0 V... V t such that (i t 0 t T 0, (ii V = V = = V t and V 0 εn, (iii for all but at most ε ( t sets {i,..., i } ( [t], the triple (Vi, V i, V i is ε-regular. A partition as given in Theorem 4 is called an (ε, t-regular partition of H. For an (ε, t-regular partition of H and d 0 we refer to Q = (V i i [t] as the family of clusters (note that the exceptional vertex set V 0 is excluded and define the cluster hypergraph K = K(ε, d, Q with vertex set [t] and {i, i, i } ( [t] being an edge if and only if (V i, V i, V i is ε-regular and d(v i, V i, V i d. In the following we show that the cluster hypergraph almost inherits the minimum vertex degree of the original hypergraph. The proof which we give for completeness is standard and can be found e.g. in [8] for the case of graphs. Proposition 5. For all γ > d > ε > 0 and all t 0 there exist T 0 and n 0 such that the following holds. Suppose H is a -uniform hypergraph on n > n 0 vertices which has vertex minimum degree δ (H ( γ ( n. Then there exists an (ε, t-regular partition Q with t 0 < t < T 0 such that the cluster hypergraph K = K(ε, d, Q has minimum vertex degree δ (K ( γ ε d ( t. Proof. Let γ > d > ε and t 0 be given. We apply the regularity lemma with ε = ε /44 and t 0 = max{t 0, 0/ε} to obtain T 0 and n 0. We set T 0 = T 0 and

12 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT n 0 = n 0. Let H be a -uniform hypergraph on n > n 0 vertices which satisfies δ(h (7/6 + γ ( n. By applying the regularity lemma we obtain an (ε, t - regular partition V 0 V... V t of V and let m = V = ( ε n/t denote the size of the partition classes. Let I = {i [t ]: V i is contained in more than ε ( t /8 non ε -regular triples} and observe that I < 8ε t /ε due to the property (iii of Theorem 4. Set V 0 = V 0 i I V i and let J = [t ]\I and t = J. We now claim that V 0 and Q = (V j j J is the desired partition. Indeed, we have T 0 > t t > t ( 8ε /ε t 0 and V 0 < ε n + 8ε n/ε εn/6. The property (iii follows directly from Theorem 4. For a contradiction, assume now that deg K (V j < ( γ ε d( t for some j J. Let x j denote the number of edges which intersect V j in exactly one vertex and each other V i, i J, in at most one vertex. Then, the assumption yields [( ( 7 t x j V j 6 + γ ε d m + ε ( t 8 ( V j n γ ε m + ε 6 n + d On the other hand, from the minimum degree of H we obtain ( ( ( ( 7 n x j V j 6 + γ Vj Vj n ( ( n 7 V j 6 + γ 4 t a contradiction. ( ] t m.. Fractional hom(m-tiling. To obtain a large M-tiling in the hypergraph H, we consider weighted homomorphisms from M into the cluster hypergraph K. To this purpose, we define the following. Definition 6. Let L be a -uniform hypergraph. A function h: V (L E(L [0, ] is called a fractional hom(m-tiling of L if ( h(v, e 0 v e, ( h(v = e E(L h(v, e, ( for every e E(L there exists a labeling of the vertices of e = uvw such that h(u, e = h(v, e h(w, e h(u, e By h min we denote the smallest non-zero value of h(v, e (and we set h min = if h 0 and the sum over all values is the weight w(h of h w(h = h(v, e. (v,e V (L E(L The allowed values of h are based on the homomorphisms from M to a single edge, hence the term hom(m-tiling. Given one such homomorphism, assign each vertex in the image the number of vertices from M mapped to it. In fact, for any such homomorphism the preimage of one vertex has size two, while the preimages of the other two vertices has size three. Consequently, for any family of homomorphisms of M into a single edge the smallest and the largest class of preimages can differ by a factor of / at most and this observation is the reason for condition ( in Definition 6. We also note the following.

13 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES Fact 7. There is a fractional hom(m-tiling h of the hypergraph M which has h min / and weight w(h = 8. Proof. Let x, x, w, y, y, w, z, and z be the vertices of M and let x x w, w y y, y y w, and w z z be the edges of M. On the edges x x w and z z w we assign the vertex weights (,, /, where the weight / is assigned to w and w. The vertex weights for edges y y w and y y w are (/, /, /, where w and w get the weight /. It is easy to see that those vertex weights give rise to a hom(m-tiling h on M with h min = / and w(h = 8. The notion hom(m-tiling is also motivated by the following proposition which shows that such a fractional hom(m-tiling in a cluster hypergraph can be converted to an integer M-tiling in the original hypergraph. Proposition 8. Let Q be an (ε, t-regular partition of a -uniform, n-vertex hypergraph H with n > ε and let K = K(ε, 6ε, Q be the corresponding cluster hypergraph. Furthermore, let h: V (K E(K [0, ] be a fractional hom(m- tiling of K with h min /. Then there exists an M-tiling of H which covers all but at most (w(h 7tε V vertices. Proof. We restrict our consideration to the subhypergraph K K consisting of the hyperedges with positive weight, i.e., e = abc K with h(a, h(b, h(c h min. For each a V (K let V a be the corresponding partition class in Q. Due to the property ( of Definition 6 we can subdivide V a (arbitrarily into a collection of pairwise disjoint sets (Ua e a e K of size Ua e = h(a, e V a. Note that every edge e = abc K corresponds to the (ε, 6ε-regular triplet (V a, V b, V c. Hence we obtain from the definition of regularity and h min / that the triplet (Ua, e Ub e, U c e is (ε, 6ε- regular. From the property ( in Definition 6 and Proposition we obtain an M-tiling of (Ua, e Ub e, U c e incorporating at least ( h(a, e + h(b, e + h(c, e 9ε V a vertices. Applying this to all hyperedges of K we obtain an M-tiling incorporating at least ( h(a, e + h(b, e + h(c, e 9ε V a ( w(h 9 K ε V a abc=e K vertices. Noting that K t (because of h min / and V a V we obtain the proposition. Owing to Proposition 8, we are given a connection between fractional hom(m- tilings of the cluster hypergraph K of H and M-tilings in H. A vertex i V (K corresponds to a class of vertices V i in the regular partition of H. The total vertex weight h(i essentially translates to the proportion of vertices of V i which can be covered by the corresponding M-tilings in H. Consequently, w(h essentially translates to the proportion of vertices covered by the corresponding M-tiling in H. This reduces our task to finding a fractional hom(m-tiling with weight greater than the number of vertices previously covered in K. The following lemma (Lemma 9, which is the main tool for the proof of Lemma, follows the idea discussed above. In the proof of Lemma we fix a maximal M-tiling in the cluster hypergraph K of the given hypergraph H. Owing to the minimum degree condition of H and Proposition 5, a typical vertex in the cluster hypergraph K will be contained in at least (7/6 + o( ( V (K hyperedges of K. We will show that a typical vertex u of K which is not covered by the

14 4 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT maximal M-tiling of K, has the property that (7/6 + o( 64 > 8 of the edges incident to u intersect some pair of copies of M from the M-tiling of K. Lemma 9 asserts that two such vertices and the pair of copies of M can be used to obtain a fractional hom(m-tiling with a weight significantly larger than 6, the number of vertices of the two copies of M. This lemma will come in handy in the proof of Lemma, where it is used to show that one can cover a higher proportion of the vertices of H than the proportion of vertices covered by the largest M-tiling in K. We consider a set of hypergraphs L 9 definied as follows: Every L L 9 consists of two (vertex disjoint copies of M, say M and M, and two additional vertices u and v such that all edges incident to u or v contain precisely one vertex from V (M and one vertex from V (M. Moreover, L satisfies the following properties for every a V (M and b V (M we have uab E(L iff vab E(L deg(u = deg(v 9. Lemma 9. For every L L 9 there exists a fractional hom(m-tiling h with h min / and w(h 6 +. The following proof of Lemma 9 is based on straightforward, but somewhat tedious case distinction. Proof. For the proof we fix the following labeling of the vertices of the two disjoint copies of M. Let V (M = {x, x, w, y, y, w, z, z } be the vertices and x x w, w y y, y y w, w z z be the edges of the first copy of M. Analogously, let V (M = {x, x, w, y, y, w, z, z } be the vertices and x x w, w y y, y y w, w z z be the edges of the other copy of M (see Figure.a. Moreover, we denote by X = {x, x }, Y = {y, y }, and Z = {z, z } and, let X, Y, and Z be defined analogously for M. Figure. Labels and case: a b, a b L with {b, b } {X, Y, Z } X x x X a x x a w w Y y y y y Y b b u v w w Z z z Z z z.a: Vertex labels of M and M in L.b: All edges are (a-edges The proof of Lemma 9 proceeds in two steps. First, we show that in any possible counterexample L, the edges incident to u and v which do not contain any vertex

15 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 5 from {w, w, w, w } form a subgraph of K,, (see Claim 0. In the second step we show that every edge contained in this subgraph of K,, forbids too many other edges incident to u and v, which will yield a contradiction to the condition deg(u = deg(v 9 of L (see Claim. We introduce the following notation to simplify later arguments. For a given L L 9 with M and M being the two copies of M, let L be set of those pairs (a, b V (M V (M such that uab E(L. We split L into L L according to { L, if {a, b} {w, w, w (a, b, w } =, L, otherwise. It will be convenient to view L and L as bipartite graphs with vertex classes V (M and V (M. We split the proof of Lemma 9 into the following two claims. Claim 0. For all L L 9 without a fractional hom(m-tiling with h min / and w(h 6 + /, we have L K,, where each of the sets X, Y, Z and X, Y, Z contains precisely one of the vertices of the K,. Claim 0 will be used in the proof of the next claim, which clearly implies Lemma 9. Claim. Let F = { a b V (M V (M : a {w, w } or b {w, w } } and for every edge ab L let F(a, b F be the set of those e F, whose appearance in L (i.e. e L implies the existence of a fractional hom(m-tiling h with h min / and w(h 6 + /. Then there is an injection f : L F such that f(a, b F(a, b for every pair ab L. Clearly, F = 8 and L F. Hence, from L + f(l = L + L 9 we derive that L and f(l must intersect. By Claim this yields the desired fractional hom(m-tiling and Lemma 9 follows. In the proofs of Claim 0 and Claim we will consider fractional hom(m- tilings h which use vertex weights of special types. In fact, for an edge e = a a a, the weights h(a, e, h(a, e, and h(a, e will be of the following forms (a h(a, e = h(a, e = h(a, e = (a h(a, e = h(a, e = h(a, e = (a h(a, e = h(a, e = h(a, e = (b h(a, e = h(a, e = and h(a, e = (b h(a, e = h(a, e = and h(a, e = (b h(a, e = h(a, e = and h(a, e = An edge that satisfies (a is called an (a-edge, etc. Note that all these types satisfy condition ( of Definition 6. Proof of Claim 0. Given L L 9 satisfying the assumptions of the claim and with the labeling from Figure.a. Observe that for any A {X, Y, Z}, the hypergraph M A contains two disjoint edges. Similarly, for every B {X, Y, Z }, M B contains two disjoint edges. First we exclude the case that there is a matching {a b, a b } of size two in L between some {a, a } = A {X, Y, Z} and some {b, b } = B {X, Y, Z }. In this case we can construct a fractional hom(m-tiling h as follows: Choose two

16 6 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Figure. Case: ab, ab L with {b, b } {X, Y, Z }.a: (a-edges w y y, w z z, and w z z, (b-edges ax u and ax v, (b- edge w y y, and (a-edge x x w..b: (a-edges w y y, w z z, and w z z, (b-edges ay u and ay v, (b- edge x x w, and (a-edge w y y. edges ua b, va b. Using these and the four disjoint edges in (M A (M B, we obtain six disjoint edges (see Figure.b. Letting all these six edges be (a- edges, we obtain a fractional hom(m-tiling h with h min = and w(h = 8. Next, we show that each vertex a A {X, Y, Z} has at most one neighbour in each B {X, Y, Z }. Assuming the contrary, let a A {X, Y, Z} and {b, b } = B {X, Y, Z }, such that ab, ab L. For symmetry reasons, we only need to consider the case B = X and B = Y. The case B = Z is symmetric to B = X. In those cases, we choose h as shown in Figure.a (B = X and Figure.b (B = Y and in either case we find a fractional hom(m-tiling h satisfying h min = / and w(h = 6 + /. Note that the cases A = Y and A = Z can be treated in the same manner since the only condition needed to define h is that M A contains two disjoint edges. To show that L is indeed contained in a K, it remains to verify that every a b, a b with {a, a } = A {X, Y, Z}, b B {X, Y, Z }, and b B {X, Y, Z } \ B guarantees the existence of a fractional hom(m-tiling h with h min / and w(h 6 + /. Again owing to the symmetry, the only cases we need to consider are B = X, B = Y (see Figure.a and B = X, B = Z (see Figure.b. In fact, the fractional hom(m-tilings h given in Figure.a and Figure.b satisfy h min / and w(h = 7. Again the cases A = Y and A = Z can be treated in the same manner. This concludes the proof of Claim 0. To complete the proof of Lemma 9 it is left to prove Claim. Proof of Claim. Before defining the injection f : L F we collect some information about F(a, b with ab L. Owing to Claim 0, we may assume without loss of generality that x, y, z and x, y, z are the vertices which span all edges of L. First we consider e = y y. As shown in Figure 4.a the appearance

17 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 7 Figure. Case: a b, a b L with {b, b } {X, Y, Z }.a: (a-edges w y y, w z z, and w z z, (b-edges a x u and a y v, and (b-edges x x w and w y y..b: (a-edges w y y and w z z, (b- edges a x u and a z v, (b-edges x x w and w z z, and (a-edges w y y and y y w. of w y L would give rise to a fractional hom(m-tiling h with h min / and w(h = 6.5. Consequently, we have w y F(y, y. For the case e = x x L, Figure 4.b, shows that x w F(x, x and by symmetry, it follows that {x w, w x } F(x, x. By applying appropriate automorphisms to M and M we immediately obtain information on F(x, z, F(z, x, and F(z, z. Indeed, we have {x w, w z } F(x, z, {w x, z w } F(z, x, {z w, w z } F(z, z. Next we consider e = y x. In this case Figure 5.a shows that y w F(y, x. Moreover, as shown in Figure 5.b we also have w x F(y, x and, consequently, we obtain {y w, w x } F(y, x. Again applying appropriate automorphisms to M and M we immediately obtain information on F(x, y, F(y, z, and F(z, y. Indeed one can show {w y, x w } F(x, y, {y w, w z } F(y, z, {w y, z w } F(z, y. Finally, we define an injection f : L F L such that f(a, b F(a, b for every pair ab L. Due to Claim 0 we have L {x, y, z } {x, y, z } and it

18 8 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Figure 4. F(y, y and F(x, x 4.a: (a-edge w z z, (a-edge y y u, (b-edges x x w, w z z, and x x w, and (b-edges w y v, y y w, and w y y. 4.b: (a-edges x x u, w y y, and w z z, (b-edges x w v and w z z, and (b-edges w y y and y y w. Figure 5. F(y, x 5.a: (a-edges y x u, x x w, and w z z, (b-edges y w v and w z z, and (b-edges w y y and y y w. 5.b: (a-edge w z z, (a-edge y x u, (b-edges x x w, w z z, and w y y, and (b-edges w x v, y y w, and x x w. follows from the discussion above that we can define f as follows f(x, x = w x, f(x, y = x w, f(x, z = x w, f(y, x = y w, f(y, y = w y, f(y, z = w z, f(z, x = w x, f(z, y = w y, f(z, z = z w.

19 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES 9 Consequently, L F L and Claim follows from L + L F 8... Proof of the M-tiling Lemma. In this section we prove Lemma. Let H be a -uniform hypergraph on n vertices. We say H has a β-deficient M-tiling if there exists a family of pairwise disjoint copies of M in H leaving at most βn vertices uncovered. Proposition. For all / > d > 0 and all β, δ > 0 the following holds. Suppose there exists an n 0 such that every -uniform hypergraph H on n > n 0 vertices with minimum vertex degree δ (H d ( n has a β-deficient M-tiling. Then every -uniform hypergraph H on n > n 0 vertices with δ (H (d δ ( n has a (β + 5 δ-deficient M-tiling. Proof. Given a -uniform hypergraph H on n > n 0 vertices with minimum degree δ (H (d δ ( n. By adding a set A of δn new vertices to H and adding all triplets to H which intersect A we obtain a new hyperpgraph H on n = n + A vertices which satisfies δ (H d ( n. Consequently, H has a β-deficient M-tiling and by removing the M-copies intersecting A, we obtain a (β + 5 δ-deficient M-tiling of H. Proof of Lemma. Let γ > 0 be given and we assume that there is an α > 0 such that for all n 0 there is a -uniform hypergraph H on n > n 0 vertices which satisfies δ (H ( γ( n but which does not contain an α-deficient M-tiling. Let α0 be the supremum of all such α and note that α 0 is bounded away from one due to Proposition. We choose ε = (γα 0 / 00. Then, by definition of α 0, there is an n 0 such that all -uniform hypergraphs H on n > n 0 vertices satisfying δ (H ( γn have an (α 0 + ε-deficient M-tiling. Hence, by Proposition all -uniform hypergraphs H on n > n 0 vertices satisfying δ (H ( γ ε( n have an (α0 + ε + 5 ε- deficient M-tiling. We will show that there exists an n (to be chosen such that all -uniform hypergraphs H on n > n vertices satisfying δ (H ( γ( n have an (α 0 ε-deficient M-tiling, contradicting the definition of α 0. To this end, we apply Proposition 5 with the constants γ, ε/, d = ε/ and t 5 = max{n 0, (ε/ } to obtain an n 5 and T 5. Let n max{n 5, n 0 } be sufficiently large and let H be an arbitrary -uniform hypergraph on n > n vertices which satisfies δ (H ( γ( n but which does not contain an α0 -deficient M-tiling. We apply Proposition 5 to H with the constants chosen above and obtain a cluster hypergraph K = K(ε/, ε/, Q on t > t 5 vertices which satisfies δ (K ( γ ε( t. Taking M to be the largest M-tiling in K we know by the definition of α 0 and by Proposition that M is an α -deficient M-tiling of K, for some α α ε. We claim that M is not (α 0 /-deficient and for a contradiction, assume the contrary. Then, from Fact 7, we know that for each M j M there is a fractional hom(m-tiling h j of M j with h j min / and weight w(hj = 8. Hence, the union of all these fractional hom(m-tiling gives rise to a fractional hom(m-tiling h of K with h min / and weight w(h 8 M t( α 0 /. By applying Proposition 8 to the fractional hom(m-tiling h (and recalling that the vertex classes V,..., V t of the regular partition has the same size, which was

20 0 ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT at least ( ε/n/t we obtain an M-tiling of H which covers at least ( ( w(h tε ε n t ( α 0 + εn vertices of H. This, however, yields a (α 0 ε deficient M-tiling of H contradicting the choice of H. Hence, M is not (α 0 /-deficient from which we conclude that X, the set of vertices in K not covered by M, has size X α 0t. (8 For a pair M i M j ( M we say the edge e K is ij-crossing if e V (Mi = e V (M j =. Claim. Let C be the set of all triples xij such that x X, M i M j ( M and there are at least 9 ij-crossing edges containing x. Then we have C γ ( t X /7. Proof. Let A be the set of those hyperedges in K which are completely contained in X and let B be the set of all the edges with exactly two vertices in X. Then it is sufficient to show that A 7 6 ( X and B 7 ( X M. (9 Indeed, assuming (9 and C γ ( t X /7 we obtain the following contradiction ( ( ( M 8 deg(x A + B + 8 X C + 64 C + M X x X [ ( 7 X X + 7 ( M 6 X M ( ( ] t 8 7 γ + M [( 7 X 6 + γ ( ( ] t 8 + M < X δ (K where in the third inequality we used ( ( t = X ( + 8 X M + 8 M. Note that the first part of (9 trivially holds since in the opposite case, using the first part of Proposition we obtain a tight path in X of length at least eight. However, this path contains a copy of M as a subhypergraph which yields a contradiction to the maximality of M. To complete the proof let us assume B > 7 there is an M M such that V (M intersects at least 7 ( X M from which we deduce that ( X edges from B. From V (M we remove the vertices which are contained in less than X edges from B. Note that there are at least four vertices, say v,..., v 4, and at least ( + ε ( X edges from B left which intersect these vertices. Hence, there exists a pair x, x such that {x, x, v i } H for all i =,..., 4. Removing all edges intersecting x, x we still have at least ( + ε/ ( X edges intersecting v,..., v 4 and we can find another pair x, x 4 disjoint from x, x with {x, x 4, v i } H for i =,..., 4. For each v i we can find another edge from B containing v i, keeping them all mutually disjoint and also disjoint from {x, x } and {x, x 4 }. This is possible since each v i is contained in more than X edges from B. This, however, yields two copies of M which contradicts the fact that M was a largest possible M-tiling.

21 MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES The set X will be used to show that there is an L L 9 such that K contains many copies of L. Claim 4. There is an element L L 9 and a family L of vertex disjoint copies of L in the cluster hypergraph K(ε/, ε/, Q such that L γα 0 t/ 75. Proof. We consider the -uniform hypergraph C (as given from Claim on the vertex set X M. Note that for fixed ij a vertex x is contained in at most 64 ijcrossing edges, thus there are at most 64 different hypergraphs with the property that x is contained in at least 9 edges which are ij-crossing. We colour each edge xij by one of the 64 colours, depending on the -partite hypergraph induced on x, M i, andm j. On the one hand, we observe that a monochromatic tight path consisting of the two edges xij, x ij C corresponds to a copy of L. On the other hand, Claim implies that there is a colour such that at least C 64 γ ( t X ( α 0γt 7 edges in C are coloured by it. Hence, by Proposition there is a tight path with α 0 γt/ 7 vertices using edges of this colour only. Note that in this tight path every three consecutive vertices contain one vertex from X and the other two vertices are from M. Thus, this path gives rise to at least α 0 γt/ 75 pairwise vertex disjoint tight paths on four vertices such that the ends are vertices from X. For any L i L we know from Lemma 9 that there is a fractional hom(m- tiling h i of L i with h i min / and weight w(hi 6 + /. Furthermore, for every M j M which is not contained in any L i L we know from Fact 7 that there is a fractional hom(m-tiling of h j of M j with h j min / and weight w(h j = 8. Hence, the union of all these fractional hom(m-tiling gives rise to a fractional hom(m-tiling h of K with h min / and weight ( w(h 6 + L + 8( M L = 8 M + L. By applying Proposition 8 to the fractional hom(m-tiling h (and recalling that the vertex classes V,..., V t of the regular partition has the same size which was at least ( ε/n/t we obtain an M-tiling of H which covers at least ( ε ( w(h tε ( ε n t ( 8 M + L tε n t vertices of H. Since M was an (α 0 +6 ε-deficient M-tiling of K, the tiling we obtained above is an (α 0 ε-deficient M-tiling of H due to the choice of ε. This, however, is a contradiction to the fact that H does not permit an (α 0 ε-deficient M-tiling..4. Proof of the path-tiling lemma. In this section we prove Lemma 0. The proof will use the following proposition which has been proven in [4] (see Lemma 0 in an even more general form, hence we omit the proof here. Proposition 5. For all d and β > 0 there exist ε > 0, integers p and m 0 such that for all m > m 0 the following holds. Suppose V = (V, V, V is an (ε, d-regular triple with V i = m for i =, and V = m. Then there there is a loose path tiling of V which consists of at most p pairwise vertex disjoint paths and which covers all but at most βm vertices of V.

MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS

MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3-unifor hypergraph on n vertices with iniu

More information

Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion. Ernie Croot. February 3, 2010 Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

More information

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Partitioning edge-coloured complete graphs into monochromatic cycles and paths arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

More information

On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

More information

On an anti-ramsey type result

On an anti-ramsey type result On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

More information

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

More information

Degree Hypergroupoids Associated with Hypergraphs

Degree Hypergroupoids Associated with Hypergraphs Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

More information

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

More information

The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

More information

Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

More information

Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

More information

8. Matchings and Factors

8. Matchings and Factors 8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

More information

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

More information

Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

More information

On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

More information

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

More information

Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

More information

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

More information

SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

More information

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

More information

Extremal Graph Theory

Extremal Graph Theory Extremal Graph Theory Instructor: Asaf Shapira Scribed by Guy Rutenberg Fall 013 Contents 1 First Lecture 1.1 Mantel s Theorem........................................ 3 1. Turán s Theorem.........................................

More information

Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10

Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

More information

1 The Line vs Point Test

1 The Line vs Point Test 6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

Collinear Points in Permutations

Collinear Points in Permutations Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for C. F. Miller Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

More information

Ramsey numbers for bipartite graphs with small bandwidth

Ramsey numbers for bipartite graphs with small bandwidth Ramsey numbers for bipartite graphs with small bandwidth Guilherme O. Mota 1,, Gábor N. Sárközy 2,, Mathias Schacht 3,, and Anusch Taraz 4, 1 Instituto de Matemática e Estatística, Universidade de São

More information

2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines 2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

Classical Analysis I

Classical Analysis I Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

More information

All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

More information

1 Formulating The Low Degree Testing Problem

1 Formulating The Low Degree Testing Problem 6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

More information

On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

More information

Game Chromatic Index of Graphs with Given Restrictions on Degrees

Game Chromatic Index of Graphs with Given Restrictions on Degrees Game Chromatic Index of Graphs with Given Restrictions on Degrees Andrew Beveridge Department of Mathematics and Computer Science Macalester College St. Paul, MN 55105 Tom Bohman, Alan Frieze, and Oleg

More information

4. Expanding dynamical systems

4. Expanding dynamical systems 4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

More information

Minimum degree condition forcing complete graph immersion

Minimum degree condition forcing complete graph immersion Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139

More information

High degree graphs contain large-star factors

High degree graphs contain large-star factors High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

More information

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,... 44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

More information

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010 CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

More information

The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

More information

The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

More information

On-line secret sharing

On-line secret sharing On-line secret sharing László Csirmaz Gábor Tardos Abstract In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified

More information

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

More information

SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

Maximum Hitting Time for Random Walks on Graphs. Graham Brightwell, Cambridge University Peter Winkler*, Emory University

Maximum Hitting Time for Random Walks on Graphs. Graham Brightwell, Cambridge University Peter Winkler*, Emory University Maximum Hitting Time for Random Walks on Graphs Graham Brightwell, Cambridge University Peter Winkler*, Emory University Abstract. For x and y vertices of a connected graph G, let T G (x, y denote the

More information

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there - 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

More information

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

More information

3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs 3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS

HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS HAMILTONIAN DEGREE SEQUENCES IN DIGRAPHS DANIELA KÜHN, DERYK OSTHUS AND ANDREW TREGLOWN Abstract. We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out- and

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

3. Equivalence Relations. Discussion

3. Equivalence Relations. Discussion 3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

More information

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Private Approximation of Clustering and Vertex Cover

Private Approximation of Clustering and Vertex Cover Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, Ben-Gurion University of the Negev Abstract. Private approximation of search

More information

Strong Ramsey Games: Drawing on an infinite board

Strong Ramsey Games: Drawing on an infinite board Strong Ramsey Games: Drawing on an infinite board Dan Hefetz Christopher Kusch Lothar Narins Alexey Pokrovskiy Clément Requilé Amir Sarid arxiv:1605.05443v2 [math.co] 25 May 2016 May 26, 2016 Abstract

More information

On the k-path cover problem for cacti

On the k-path cover problem for cacti On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Non-Separable Detachments of Graphs

Non-Separable Detachments of Graphs Egerváry Research Group on Combinatorial Optimization Technical reports TR-2001-12. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

More information

Introduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes.

Introduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes. 68 Part II. Combinatorial Models can require a number of spin flips that is exponential in N (A. Haken et al. ca. 1989), and that one can in fact embed arbitrary computations in the dynamics (Orponen 1995).

More information

A 2-factor in which each cycle has long length in claw-free graphs

A 2-factor in which each cycle has long length in claw-free graphs A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

More information

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES 136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

More information

Tilings of the sphere with right triangles III: the asymptotically obtuse families

Tilings of the sphere with right triangles III: the asymptotically obtuse families Tilings of the sphere with right triangles III: the asymptotically obtuse families Robert J. MacG. Dawson Department of Mathematics and Computing Science Saint Mary s University Halifax, Nova Scotia, Canada

More information

Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

More information

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

arxiv:1112.0829v1 [math.pr] 5 Dec 2011 How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

More information

About the inverse football pool problem for 9 games 1

About the inverse football pool problem for 9 games 1 Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

More information

PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Problem Set I: Preferences, W.A.R.P., consumer choice

Problem Set I: Preferences, W.A.R.P., consumer choice Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y

More information

Cycles and clique-minors in expanders

Cycles and clique-minors in expanders Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor

More information

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

More information

BOUNDARY EDGE DOMINATION IN GRAPHS

BOUNDARY EDGE DOMINATION IN GRAPHS BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

More information

8.1 Min Degree Spanning Tree

8.1 Min Degree Spanning Tree CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

More information

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

More information

Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

More information

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Suil O, Douglas B. West November 9, 2008; revised June 2, 2009 Abstract A balloon in a graph G is a maximal 2-edge-connected

More information