MIXDECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia


 Gyles Davidson
 1 years ago
 Views:
Transcription
1 GLASNIK MATEMATIČKI Vol. 38(58)(2003), MIXDECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia Abstract. We estimate the values, for each k, of the smallest n such that K n can be mixdecomposed into k undirected factors of diameter 3 and one directed factor of diameter 2. We find the asymptotic value of ratio of n and k, when k tends to infinity and generalize this result for mixdecompositions into p directed factors of diameter 2 and k undirected factors of diameter 3.. Introduction In this paper, we solve the following problem. There is a system of n communication devices and there are k + communication networks. There are k communication networks that allow twoway communication (the link of this network connecting two devices allows each of them to receive and send information) and the remaining communication network allows only oneway communication (the link of this network connecting two devices allows one of them only to send information and the other only to receive them). The following requirements are given: ) Every pair of different devices can be connected only by a single link. 2) For every pair of different devices and for every twoway communication network, there has to be a path of at most 3 links through which they can communicate. 3) For every pair of different devices a and b there has to be a path of at most 2 links of the oneway communication network through which a can send information to b and a path of at most 2 links of the oneway communication network through which b can send information to a Mathematics Subject Classification. 05C70. Key words and phrases. Decomposition, mixdecomposition, factor, graph. 2
2 22 D. VUKIČEVIĆ Of course, it is not always possible to meet these requirements. Take for instance the case n = 3 and k =. In this paper, we want to estimate, for each k, the smallest n such that this is possible This problem is closely connected to wellknown problems of decompositions of graphs. Decompositions of graphs into factors with given diameters have been extensively studied. The problem of decomposition of the factors of equal diameters, where diameter of each factor is at least three has been solved in [4]. The majority of the papers written about decomposition of the graphs are written about decompositions into the factors of diameter two. Denote by f (k) the smallest natural number so that complete graph with n vertices can be decomposed into k factors of diameter 2. In [5], it was proved that f (k) 7k. Then in [3] this was improved to f (k) 6k. In [7], it was further proved that this upper bound is quite close to the exact value of f (k) since f (k) 6k 7, k 664 and in [8] the correct value of f (k) was given for large values of k, to be more specific it was proved that f (k) = 6k, k 0 7. Therefore, the most interesting problem of decompositions of graphs are decompositions into factors of small diameters. The problem most closely related to ours, i.e. the problem of decompositions of complete graph into factors of diameter 2 and 3 was treated in [6]. We obtain few results similar to the results of that paper, but these constructions will be somewhat more complicated, since in this paper we have to deal with directed factors of diameter 2. Also, the organization of the system of communication networks presented here is somewhat better than one given in [6], because in this system the privileged communication network is much faster; here links of privileged network allow only oneway communication, which is much faster then twoway communication. 2. Basic definitions Let G be undirected (resp. directed graph). By V (G), we denote set of its vertices, by v (G) number of its vertices, by E (G) set of its edges (resp. directed edges) and by e (G) the number of its edges (resp. directed edges). By d G (x, y) we denote the length of the shortest path connecting x and y (resp. the length of the shortest path from x to y). We also denote diam G = max {d G (x, y) : x, y V (G)}.
3 MIXDECOMPOSITON OF THE COMPLETE GRAPH 23 We shall slightly abuse the word subgraph (resp. supergraph) by saying that A is a subgraph of B (resp. B is a supergraph of A) if B contains a subgraph isomorphic to A. For undirected graph G, we denote by d G (x) degree of vertex x, by δ (G) minimal degree of the graph and by (G) maximal degree of the graph. For a directed graph G we denote by d + (x) outdegree of vertex x and by d (x) indegree of vertex x. By + (G), we denote maximal outdegree of graph G and by (G) maximal indegree of graph G. By δ + (G) we denote minimal outdegree of graph G and by δ (G) minimal indegree of graph G. We say that vertex x is naccessiblefrom (resp. naccessibleto) y if there is a path from y to x (resp. from x to y) of length at most n, and for any set of vertices A, we say that A is naccessiblefrom (resp. naccessibleto) x if each vertex in A is naccessiblefrom (resp. naccessibleto) x. If (x, y) E (G), we say that y is outneighbor of x and that x is inneighbor of y. Let D be a directed graph. Denote by D a graph such that V ( D ) = V (D) and We also define xy E ( D ) ((x, y) E (D) (y, x) E (D)) Definition 2.. Let G be undirected graph. We say that G is mixdecomposed into undirected factors F, F 2,..., F k and directed factors D, D 2,..., D p if V (F ) = V (F 2 ) = = V (F k ) = V (D ) = V (D 2 ) = = V (D p ) = V (G) and for each edge {a, b} E (G) exactly one of the following is true: ) {a, b} E (F ) 2) {a, b} E (F 2 ). k) {a, b} E (F k ) k+) (a, b) E (D ) k+2) (a, b) E (D 2 ). k+p) (a, b) E (D p ) k+p+) (b, a) E (D ) k+p+2) (b, a) E (D 2 ). k+2p) (b, a) E (D p ) Definition 2.2. Let G be an undirected graph. Mixcomplement of G is any directed graph G such that V (G) = V (G ) and that for each a, b V (G ) exactly one of the following statements is true:
4 24 D. VUKIČEVIĆ ) {a, b} V (G) 2) (a, b) E (G ) 3) (b, a) E (G ). Definition 2.3. Let G be a directed graph. Mixcomplement of G is any undirected graph G such that V (G) = V (G ) and such that for each a, b V (G ) exactly one of the following statements is true: ) {a, b} V (G ) 2) (a, b) E (G) 3) (b, a) E (G). We define the function φ : N N by φ (k) = n if and only if n is the smallest natural number such that K n can be mixdecomposed into k undirected factors of diameter 3 and one directed factor of diameter 2. Therefore, we have to estimate the values of the function φ. 3. The value of φ () It can be easily, but tediously proved that (if the reader is interested in the details of this and other missing proofs, please send me a mail): Lemma 3.. There is no mixcomplement D of cycle with 7 vertices such that diam D 2. Now, we prove Lemma 3.2. φ () > 7. Proof. Suppose to the contrary, that it is possible to mixdecompose K n, n 7, into undirected factor F of diameter 2 and directed factor D of diameter 3. It is easy to see that δ + (D) 2 and δ (D) 2, because otherwise it would be δ (F ) = 0, which is impossible. If n 4, then this is impossible. If n = 5, then (F ) = 0. If n = 6, then (F ), hence F is disconnected. If n = 7, then (F ) 2 and diam F 3, so F is a cycle, but this is in contradiction with the previous lemma. In all the cases we have obtained a contradiction, therefore φ () > 7. Let us prove that φ () > 8. Denote by G 8,, G 8,2 and G 8,3 the graphs on Figures, 2 and 3. By a simple analysis, we can prove: Lemma 3.3. Let G be a graph such that v (G) = 8 and diam G 3, (G) 3 and δ (G) =, then G is a supergraph of one of the graphs G 8,, G 8,2 and G 8,3. Simple, but tedious analysis shows that: Lemma 3.4. There is no directed graph D such that diam D 2 and D is mixcomplement of G 8,.
5 MIXDECOMPOSITON OF THE COMPLETE GRAPH 25 v v 2 v 3 v 4 v 5 v 7 v 6 v 8 Figure v v 2 v 3 v 4 v 5 v 7 v 6 v 8 Figure 2 v v 2 v 3 v 4 v 5 v 7 v 6 v 8 Figure 3 Lemma 3.5. There is no directed graph D such that diam D 2 and D is mixcomplement of G 8,2.
6 26 D. VUKIČEVIĆ Lemma 3.6. There is no directed graph D such that diam D 2 and D is mixcomplement of G 8,3. From the last four lemmas, it follows. Lemma 3.7. Let K 8 be mixdecomposed into undirected factor F of diameter at most 3 and directed factor D of diameter at most 2; then δ (F ) 2. Let us prove Lemma 3.8. Let D be directed graph with 8 vertices of diameter 2 such that δ + (D) 2, δ (D) 2 and sum of indegree and outdegree for at most 6 vertices is 5 and for the remaining vertices is 4, then exactly three vertices have indegree 3 and there is at most one edge between these 3 vertices, and also exactly three vertices have outdegree 3, and there is at most one edge between these vertices. Proof. First let us prove that each vertex of outdegree 2 has at least one outneighbor of outdegree 3. Suppose, to the contrary, that there is a vertex x of outdegree 2 such that its both outneighbors have outdegree 2. In that case, there are only 7 vertices that are 2accessiblefrom x and that is a contradiction. Now let us prove that there are at least 3 vertices with outdegree 3. Suppose to the contrary, that there are at most 2 vertices with outdegree 3. Each of these vertices can be outneighbor of at most two vertices. Since there are at least 6 vertices of outdegree 2, we have obtained a contradiction, hence there have to be at least 3 vertices of outdegree 3. We can prove analogously that there are at least 3 vertices with indegree 3, hence there are exactly three vertices with in degree 3 and 3 vertices with outdegree 3. At this point, we prove that there cannot be two edges between three vertices of outdegree three. Suppose to the contrary, that this is possible, but then only four vertices of outdegree two can have a neighbor of outdegree 3. Since, there are 5 vertices of outdegree 2, this is a contradiction. Completely analogously, it can be proved that there is at most one edge between 3 vertices that have indegree 3. Denote by G 8,4, G 8,5 and G 8,6 respectively the following graphs Figure 4
7 MIXDECOMPOSITON OF THE COMPLETE GRAPH 27 Figure 5 Figure 6 A simple analysis shows that: Lemma 3.9. Every graph G with 8 vertices such that diam G 3, δ (G) = 2, (G) 3 and at most two vertices have a degree larger than 2, is isomorphic to G 8,4, G 8,5 or G 8,6. Simple check shows that: Lemma 3.0. In complement of G 8,4 six vertices of degree 5 cannot be divided in two triples in such a way that each triple contains only one edge. Lemma 3.. In complement of G 8,5 six vertices of degree 5 cannot be divided in two triples in such a way that each triple contains only one edge. Lemma 3.2. In complement of G 8,6 six vertices of degree 5 cannot be divided in two triples in such a way that each triple contains only one edge. Combining the last five lemmas we get: Lemma 3.3. K 8 can not be mixdecomposed into undirected factor F of diameter 3 and directed factor F 2 of diameter 2, such that δ (F 2 ) 2. The last lemma and Lemma 3.7 yield the following proposition: Proposition 3.4. φ () > 8. From the last Proposition and the following decomposition of K 9 (on this sketch we draw only the directed edges of directed factor D and all the missing edges are edges of undirected factor F ): it follows that Theorem 3.5. φ () = 9.
8 28 D. VUKIČEVIĆ Figure 7 4. Estimates of φ (k) for small values of k We start with several lemmas: Lemma 4.. If there is a directed graph D of diameter 2, such that D = K n, then there is a directed graph D such that diam D = 2 and D = K n+2. Proof. Denote vertices of D by v, v 2,..., v n. We explicitly construct D. Its vertices are v,..., v n+2 and its edges are: ) (x, y) such that (x, y) E (D) 2) (x, v i ) such that n + i n + 2, (x, v n ) E (D) 3) (v i, x) such that n + i n + 2, (v n, x) E (D) 4) (v n, v n+ ), (v n+, v n+2 ), (v n+2, v n ). Lemma 4.2. There is a directed graph D (2, 3) of diameter 2 such that D (2, 3) = K 3, and there is directed graph D (2, 6) of diameter 2 such that D (2, 3) = K 6. Proof. Graphs with the required properties are given by the following sketches: Figure 8
9 MIXDECOMPOSITON OF THE COMPLETE GRAPH 29 Figure 9 The last two Lemmas yield that there are graphs D (2, n) of diameter 2 such that D (2, n) = K n for each n, 2, 4. Denote by D a mixeddecomposition of the K 0 into undirected factor F D and directed factor D D given by the following sketch (on the following sketch only the edges of the factor D D are drawn and the missing edges are edges of the factor F D ): x x 2 x 3 a a 7 a 2 a 6 a 3 a 4 a 5 Figure 0 It can be easily seen that diam F D = 3 and that diam D D = 2. Let us prove that: Lemma 4.3. φ (k) 3k + 7, k, 2, 4. Proof. We explicitly give a mixeddecomposition of K 3k+7 into undirected factors F,..., F k and directed factor D with a required properties. Denote k V (K 3k+7 ) = A i= B i A = {a, a 2,..., a 6 }, B i = {b i,, b i,2, b i,3 }.
10 220 D. VUKIČEVIĆ Edges of F i, for each i k, are: ) b i,j a k such that x j a k E (F D ) 2) b i, b l,2, b i,2 b l,3, b i,3 b l,, l k, i l 3) b i, b i,2, b i,2 b i,3, b i,3 b i,. Since b i,, b i,2, b i,3 are all adjacent in F i and each vertex is adjacent to at least one of them, it follows that diam F i = 3. Denote by g i : {b,i, b 2,i,..., b k,i } V (D (2, k)), i =,..., 3 any bijections. Directed edges of D are: ) (a i, a j ) such that (a i, a j ) V (D D ) 2) (b i,j, a l ) such that (x j, a l ) V (D D ) 3) (a l, b i,j ) such that (a l, x j ) V (D D ) 4) (b i,j, b l,j ) such that (g j (b i,j ), g j (b l,j )) V (D D ). Let us prove that diam D 2. We need to show that d (x, y) 2 for each x, y V (D). Distinguish three cases: ) x, y A or x A, y B i, i k or x B i, y A, i k or x, y B i, i k. Note that there is a subgraph of D isomorphic to D D that contains x and y, hence the claim follows. 2) x = b i,j, y = b l,m, i l, j m, i, j, l, m k. There is a vertex a o, o 7 such that (x j, a o ),(a o, x m ) E (D D ). Therefore (b i,j, a o ), (a o, b l,m ) E (D D ). 3) x = b i,j, y = b l,j. There is a path of length at most 2 that consists of directed edges listed in 4). We have exhausted all the cases and we have proved our claim. Lemma 4.4. φ (2) 4. Proof. We explicitly give a decomposition of K 4 into undirected factors F and F 2 of diameter 3 and directed factor D of diameter 2 by the following table T :
11 MIXDECOMPOSITON OF THE COMPLETE GRAPH where T ij = k denotes ij F k, k 2; T ij = 3 denotes (i, j) E (D) and T ij = 4 denotes (j, i) E (D). Lemma 4.5. φ (4) 20. Proof. We explicitly give a decomposition of K 20 into undirected factors F, F 2, F 3 and F 4 of diameter 3 and directed factor D of diameter 2 by the following table T : where T ij = k denotes ij F k, k 4; T ij = 5 denotes (i, j) E (D) and T ij = 6 denotes (j, i) E (D). Lemma 4.6. Let D be a directed graph with n vertices of diameter 2 such that δ + (D) 2. Then D has at least 3n 7 edges. Proof. If δ + (D) 3, the claim is trivial. If not, there is a vertex x with exactly 2 outneighbours, say y and z. Since diam D 2, it follows that d + (y) + d + (z) n 3. Therefore we have v V (D) d+ (v) 3n 7. The last Lemma yields Lemma 4.7. φ (k) 2k+7+ (2k+7)
12 222 D. VUKIČEVIĆ Proof. Suppose that K n can be decomposed into k undirected factors each of diameter at most 3 and one directed factor of diameter 2. Each of undirected factors have at least n edges and directed graph by last Lemma has at least 3n 7 directed edges. Therefore, hence k n + 3n 7 n (n ) 2 n 2 (2k + 7) n + 4 0, 2k (2k + 7) 2 56 n 2. So far we have shown that Theorem , k = 2k+7+ (2k+7) , k 2 φ (k) 3k + 6 k = 3k + 8 k = 2, 4 3k + 7 k, 2, Estimates of φ (k) for large values of k The upper bounds of the last theorem are quite good for small values of k, but they are bad for large values of k. Note that from the last theorem, it follows only We shall prove that 2 lim φ (k) k lim φ (k) k 3. φ (k) lim = 2. k k This is a result analog to the result given in [6]. The techniques of proving this are similar to those of [6], but somewhat more complicated. Lemma 5.. Let t, k and q be such natural numbers that [ ( ) ( ) 2q + 2k + 4 k + 4t 2 2q + 2k + 4 k + 4t 3 ] 2k (2k ) ( ) q 3 < 4 then there is a directed graph D such that its vertices can be divided into five pairwise disjoint sets C, C 2, C 3, C 4 and C 5 in such a way that: ) C = q 2) C 2 = q 3) C 3 = k
13 MIXDECOMPOSITON OF THE COMPLETE GRAPH 223 4) C 4 = k 5) C 5 = 4 k + 4t 2 6) For any two vertices x X and y Y either (x, y) E (D ) or (y, x) E (D ), where {X, Y } {{C, C 2 }, {C, C 4 }, {C, C 5 }, {C 2, C 3 }, {C 2, C 5 }, {C }, {C 2 }} 7) For any two vertices x X and y Y neither (x, y) E (D ) nor (y, x) E (D ), where {X, Y } {{C 3 }, {C 4 }, {C 5 }, {C, C 3 }, {C 2, C 4 }, {C 3, C 4 }, {C 3, C 5 }, {C 4, C 5 }} 8) For each pair of vertices (x, y) (V (D ) V (D )) \ ((C 3 C 4 ) (C 3 C 4 )). we have d D (x, y) 2. Proof. Let D be a random graph such that V (D ) = C C 2 C 3 C 4 C 5 such that ) For any two vertices x X and y Y either (x, y) E (D ) or (y, x) E (D ), where {X, Y } {{C, C 2 }, {C, C 4 }, {C, C 5 }, {C 2, C 3 }, {C 2, C 5 }, {C }, {C 2 }} and the probability of each direction is 2. 2) For any two vertices x X and y Y neither (x, y) E (D ) nor (y, x) E (D ), where {X, Y } {{C 3 }, {C 4 }, {C 5 }, {C, C 3 }, {C 2, C 4 }, {C 3, C 4 }, {C 3, C 5 }, {C 4, C 5 }} Denote the following condition by ( ): For each pair of vertices (x, y) (V (D ) V (D )) \ ((C 3 C 4 ) (C 3 C 4 )). we have d D (x, y) 2. Let us calculate probability that D does not satisfy ( ). First we estimate the probability prob (x, y) that d D (x, y) > 2, where x y and (x, y) (V (D ) V (D )) \ ((C 3 C 4 ) (C 3 C 4 )).
14 224 D. VUKIČEVIĆ The following 2 cases, described in the following table may occur: x y prob (x, y) C C 2 ( 3 4 C C 2 2 ( 3 4 ( C C C C 4 2 ( 3 4 C C 5 2 ( 3 4 C 2 C 2 ( 3 4 C 2 C 2 2 ( 3 4 C 2 C 3 2 ( 3 ( 4 C 2 C C 2 C 5 2 ( 3 ( 4 C 3 C 3 4 C 3 C 2 2 ( 3 ( 4 C 3 C C 4 C 2 ( 3 ( 4 C 4 C 3 2 ( 4 C 4 C C 5 C 2 ( 3 4 C 5 C 2 2 ( 3 ( 4 C 5 C 3 3 ( 4 C 5 C 3 4 ( 4 C 5 C ) 2q+k+4 k +4t 4 ) 2q+4 k +4t 4 ) q ) q ) 2q ) 2q+4 k +4t 4 ) 2q+k+4 k +4t 4 ) q ) q ) 2q ) q ) q ) q ) q ) q ) q ) 2q ) 2q ) q ) q ) 2q Thus, prob (x, y) ( 3 4) q for any two different vertices x, y such that (x, y) (V (D ) V (D )) \ ((C 3 C 4 ) (C 3 C 4 )). Therefore, a probability that D does not satisfy ( ) is less then [ ( ) ( ) 2q + 2k + 4 k + 4t 2 2q + 2k + 4 k + 4t 3 ] 2k (2k ) ( ) q 3 4 Since the last expression is less than, there is a graph with the required properties. Now we can prove:
15 MIXDECOMPOSITON OF THE COMPLETE GRAPH 225 k Theorem 5.2. For any k N, we have φ (k) 2k t 2 + 2q where t is the least natural number such that ( ) 2t k. t and q is the least natural number such that [ ( ) 2q + 2k + 4 k + 4t 2 ] 2k (2k ) ( 2q + 2k + 4 ( ) q 3 <. 4 ) k + 4t 3 k Proof. We will construct a decomposition of K n, n = 2k t 2 + 2q, into undirected factors F, F 2,..., F k and directed factor F such that diam F = 2 and diam F i = 3, i k. Let V (K n ) = L D W Z U U A B B C, where } L = {l,..., l k }, D = {d,..., d k }, W = {w,..., w k, } } { } Z = {z,..., z k, U = {u,..., u k, U = u,..., u k, A = {a,..., a q }, B = {b,..., b 2t }, B = { b,..., b 2t }, C = {c,..., c q }. Let B be the set of all subsets of t elements of the set {, 2,..., 2t }. Let f be any injection f : {,..., k} B. Let us notice that for each j {,..., k} there are unique numbers q j and r j so that j = (q j ) k + r j, q j k, r j k. The edges of the factor F i, i k are ) l i d i 2) l i l j, j < i 3) d i l j, i < j k 4) l i d j, i < j k 5) d i d j, j < i 6) l i a j, j q 7) d i c j, j q 8) l i b j, l i b j, j f (i) 9) d i b j, d i b j, j {, 2,..., 2t } \ f (i) k 0) l i w j, j
16 226 D. VUKIČEVIĆ k ) d i z j, j 2) w qi u ri, z qi u ri, w qi u r i, z qi u k r i 3) d i u j, d i u j, j, j r i. In each factor F i, i k all vertices are adjacent to either l i or d i, except u ri and u r i each of which is connected by a path of length 2 to both, l i and d i. Also, l i and d i are adjacent, and there is a path of length 2 which connects u ri and u r i, hence we have diam F i = 3, i k. Let D be a digraph with a properties required in the previous lemma and let g : V (K n ) V (D ) be a bijection such that Directed edges of F are: g (A) = C g (C) = C 2 g (L) = C 3 g (D) = C 4 g (B B V Z U U ) = C 5 ) (x, y) such that (g (x), g (y)) E (D ) 2) (l i, u ri ), (u ri, d i ), ( ) ( ) d i, u r i, u ri, l i 3) (b j, d i ), ( d i, b j), j f (i) 4) (l i, b j ), ( b j, l i), j {, 2,..., t } \ f (i) 5) Edges in ( k { xy : [(x, y) is edge of D listed in )5)] E (K n ) \ E (F i ) [(y, x) is edge of D listed in )5)] i= with an arbitrary orientations. It remains to prove that diam F 2, i.e. that for arbitrary x, y, we have d (x, y) 2. Distinguish 5 cases: ) x = l i, y = d i, i k. There is a path l i u ri d i. 2) x = d i, y = l i, i k. There is a path d i u r i l i. 3) x = l i, y = d j, i j, i, j k. Since f is a bijection, there is m f (j) \ f (i) and therefore, there is a path l i b m d j. 4) x = d i, y = l j, i j, i, j k. Since f is a bijection, there is m f (i) \ f (j) and therefore, there is a path d i b ml j. } )
17 MIXDECOMPOSITON OF THE COMPLETE GRAPH 227 5) x V (K n ) \ (L D) or y V (K n ) \ (L D). There is a path of length at most 2 from x to y consisting of edges listed in ). All the cases are exhausted and the claim is proved. From the last theorem, it easily follows Corollary 5.3. lim k φ(k) k = 2. Proof. Let k N be sufficiently large. Let us find upper and lower bounds for φ (k). We have ( ) φ (k) k (φ (k) ) k φ (k) φ (k) 2k. 2 2 Let us notice that, for sufficiently large k, we have thus t therefore q ( k 2 ) k k, k. Also, for sufficiently large k, we have [ ( ) ( ) 2q + 2k + 8 k 2 2q + 2k + 8 k 3 k. It follows that ] 2k (2k ) ( ) 2k φ (k) 2k + 0 k + which proves the claim. lim k 2 lim k ( ) 3 k <, 4 2 φ (k) k k k. ( ) ( φ (k) lim ), k k k k Let us generalize the last corollary. By a simple probabilistic argument (similarly as in Lemma 5.), we show that: Lemma 5.4. Let p, r, r and k be natural numbers such that ( ( ) ) 2 r p [(r + 2r + 2k) (r + 2r + 2k ) 2k (2k )] <. 2p There is a digraph D r,r such that its vertices can be divided into five pairwise disjoint sets C (r,r ),, C (r,r ),2, C (r,r ),3, C (r,r ),4, and C (r,r ),5 in such way that: ) C(r,r ), = r
18 228 D. VUKIČEVIĆ 2) C(r,r ),2 = r 3) C (r,r ),3 = k 4) C(r,r ),2 = k 5) C(r,r ),2 = r 6) For any two vertices x X and y Y either (x, y) E (D r,r ) or (y, x) E (D r,r ), where { { } { } { } } C(r,r {X, Y } { ),, C (r,r ),2 } {, C(r,r ),, C (r,r ),4 } {, C(r,r ),, } C { (r,r ),5, C(r,r ),2, C (r,r ),3, C(r,r ),2, C (r,r ),5, C(r,r ),, C(r,r ),2} 7) For any two vertices x X and y Y neither (x, y) E (D r,r ) nor (y, x) E (D r,r ), where { } { } { } C(r,r ),, C { (r,r ),3, C(r,r ),2, } C { (r,r ),4, C(r,r ),3, } C (r,r ),4, {X, Y } C(r,r ),3, C (r,r } { ),5, C(r,r } { ),4, C (r,r }),5,, C(r,r ),4, C(r,r ),5, { C(r,r ),3 which can be decomposed into p factors P (r,r ),,..., P (r,r ),p such that 8) For each pair of vertices (x, y) (V (D r,r ) V (D r,r ))\ (( C (r,r ),3 C (r,r ),4) ( C(r,r ),3 C (r,r ),4)). and each factor P (r,r ),i, i k, we have d P(r,r ),i, (x, y) 2. Proof. Let D r,r be a random digraph such that its vertices can be decomposed into five pairwise disjoint sets C (r,r ),, C (r,r ),2, C (r,r ),3, C (r,r ),4, and C (r,r ),5 in such way that: ) For any two vertices x X and y Y either (x, y) E ( ) D r,r or (y, x) E ( ) D r,r, where { { } { } { } } C(r,r {X, Y } { ),, C (r,r ),2} {, C(r,r ),, C (r,r ),4} {, C(r,r ),, } C { (r,r ),5, C(r,r ),2, C (r,r ),3, C(r,r ),2, C (r,r ),5, C(r,r ),, C(r,r ),2} and probability of each direction is 2. 2) For any two vertices x X and y Y neither (x, y) E ( ) D r,r nor (y, x) E ( ) D r,r, where {X, Y } { } { } { } C(r,r ),, C { (r,r ),3, C(r,r ),2, } C { (r,r ),4, C(r,r ),3, } C (r,r ),4, C(r,r { ),3, C (r,r } {),5, C(r,r } {),4, C (r,r }),5, C(r,r ),3, C(r,r ),4, C(r,r ),5, Let P (r,r ),,..., P (r,r ),p be a random decomposition of random digraph (each directed edge of (x, y) has a probability p to be a directed edge of, for every i {,..., p}). Denote by ( ) the following condition: P (r,r ),i For each pair of vertices (x, y) ( V ( D r,r ) V ( D r,r )) \ (( C(r,r ),3 C (r,r ),4) ( C(r,r ),3 C (r,r ),4)). and each factorp (r,r ),i, i k, we have d P(r,r ),i, (x, y) 2.
19 MIXDECOMPOSITON OF THE COMPLETE GRAPH 229 Let us calculate the probability that D r,r does not satisfy ( ). First, we estimate the probability prob (x, y, i) that d P (x, y) > 2, where (x, y) (r,r ),i (V (D ) V (D )) \ (( ( C (r,r ),3 C (r,r ),4) C(r,r ),3 C (r,r ),4)) and i p. Analogously as in the Lemma 5., we solve this problem by observing 2 possible cases. We get ( ( ) ) 2 r prob (x, y, i). 2p for each (x, y) ( V ( D r,r ) V ( D r,r )) \ (( C(r,r ),3 C (r,r ),4) ( C(r,r ),3 C (r,r ),4)). Therefore, a probability that D r,r does not satisfy ( ) is at most ( ( ) ) 2 r p [(r + 2r + 2k) (r + 2r + 2k ) 2k (2k )]. 2p Since the last expression is less than, there is a graph with the required properties. We can now prove: Theorem 5.5. Let p and k be any natural numbers. Then K n can be mixeddecomposed into k undirected factors F, F 2,..., F k of diameter 3 and p directed factors D, D 2,..., D p of diameter 2 where n = 2k + 2r + 2p (2t ) + (2p + 2) k, where t is the smallest integer such that ( ) 2t k t and r is the smallest integer such that p [(r + 2r + 2k) (r + 2r + 2k ) 2k (2k )] ( ( ) ) 2 r <. 2p where r = 2p (2t ) + (2p + 2) k. Proof. Denote p p p p V (K n ) = L R A B C α E α X Y U α V α, α= α= α= α=
20 230 D. VUKIČEVIĆ L = {l,..., l k }, R = {r,..., r k }, A = {a,... a r }, B = {b,..., b r }, } } X = {x,..., x k, Y = {y,..., y k, and for, each α =,..., p, denote C α = { c α,..., 2t } cα, Eα = { e α,..., 2t } eα, { } { } U α = u α,..., uα k, V α = v α,..., vα k. Let us notice, as in the Theorem 5.2 that for each j {,..., k} there are unique numbers Q j and R j such that j = (Q j ) k + R j, Q j k, R j k. Also, as in the Theorem 5.2, let B be the set of all subsets of t elements of the set {, 2,..., 2t } and let f be any injection f : {,..., k} B. We explicitly give a mixeddecomposition with the required properties. The edges of F i, i =,..., k are: ) l i r i 2) l i l j, r i r j, i < j k 3) l i r j, r i l j, j < i 4) l i a j, j =,..., r 5) r i b j, j =,..., r 6) l i c α j, l ie α j, j f (i), α =,..., p 7) r i c α j, r ie α j, j {, 2,..., 2t } \ f (i), α =,..., p k 8) l i x j, r i y j, j =,..., 9) x Qi u α R i, y Qi u α R i, x Qi v α R i, y Qi v α R i, α =,..., p 0) l i u α j, r iv α j, j k, j R i, α p It can be easily checked that diam F i = 3, for each i =,..., k. Let g : V (K n ) V (D r,r ) be any bijection such that g ( X Y g (A) = C (r,r ), g (B) = C (r,r ),2 g (L) = C (r,r ),3 g (R) = C (r,r ),4 ) p (C α E α U α V α ) = C (r,r ),5 α=
21 MIXDECOMPOSITON OF THE COMPLETE GRAPH 23 Directed edges of D α, α =,..., p are: ) (x, y) such that ( ) g (x), g (x) P (r,r ),α 2) ( ) ( ) ( ) ( ) l i, u α R i, u α Ri, r i, ri, vr α i, v α Ri, l i 3) ( ) ( ) l i, c α j, e α j, l i, j {, 2,..., 2t } \ f (i), i =,..., k 4) ( c α j, r ( ) i), ri, e α j, j f (i), i =,..., k. Edges in k E (F i ) p E ( D α ) E (K n ) \ { i= α=2 } xy : [(x, y) is edge of D listed in )4)] [(y, x) is edge of D listed in )4)] are directed edges of D with an arbitrary orientations. Let us prove that diam D α 2, α =,..., p, i.e. that d Dα (x, y) 2, for each x, y V (K n ). Distinguish 5 cases: ) x = l i, y = r i. There is a path l i u α R i r i. 2) x = r i, y = l i. There is a path r i v α R i l i. 3) x = l i, y = r j, i j. Since f is a bijection, there is m f (j) \ f (i) and therefore, there is a path l i c α m r j. 4) x = r i, y = l j, i j. Since f is a bijection, there is m f (i) \ f (j) and therefore, there is a path r i e α m l j. 5) x V (K n ) \ (L R) or y V (K n ) \ (L D). There is a path of length at most 2 from x to y consisting of edges listed in ). All the cases are exhausted and the claim is proved. Corollary 5.6. Let p be a fixed natural number and let Φ (p, k) be the smallest natural number such that K Φ(p,k) can be decomposed into p directed factors of diameter 2 and k undirected factors of diameter 3. Then Φ (p, k) lim = 2. k k Proof. From the last theorem, it follows that Φ (p, k) 2k + 2r + 2p (2t ) + (2p + 2) where t is the smallest integer such that ( ) 2t k t k,
22 232 D. VUKIČEVIĆ and r is the smallest integer such that p [(r + 2r + 2k) (r + 2r + 2k ) 2k (2k )] ( where r = 2p (2t ) + (2p + 2) k. k Note that, for sufficiently large k, we have t and r φ (k) Φ (p, k) 2k + (6p + 4) k φ (k) lim k k which proves the claim. ( ) ) 2 r <. 2p k, hence k Φ (p, k) (6p + 4) lim lim 2 +, k k k k Acknowledgements. The author would like to thank D. Veljan, D. Svrtan, G. Igally, and G. Duplančić for useful advises and help. References [] B. Bollobás: Extremal Graph Theory, Academic Press, London, 978. [2] J. Bosák: Decompositions of Graphs, Kluwer Academic Publishers, Dordecht, 989. [3] J. Bosák: Disjoint Factors of Diameter Two in Complete Graphs, Journal Of Combinatorial Theory (B) 6 (974), [4] D. Palumbíny: On Decompositions of Complete Graphs into Factors with Equal Diameters, Bollettino U.M.I. (4) 7 (973), [5] N. Sauer: On Factorization of Complete Graphs into Factors of Diameter Two, Journal of Combinatorial Theory 9 (970), [6] D. Vukičević: Decomposition of Complete Graph into Factors of Diameter Two and Three, Discussiones Mathematicae Graph Theory 23 (2003), [7] Š. Znám: Decomposition of Complete Graphs into Factors of Diameter Two, Mathematica Slovaca 30 (980), [8] Š. Znám: On a Conjecture of a Bollobás and Bosák, Journal of Graph Theory 6 (962), D. Vukičević Department of Mathematics University of Split Teslina 2, 2000 Split Croatia Received: Revised:
SCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationSHARP 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 information2.3 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 informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More informationThe 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 informationMidterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
More informationDiscrete 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 informationA 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 informationGRAPH THEORY LECTURE 4: TREES
GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection
More informationThe 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 kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationSmooth functions statistics
Smooth functions statistics V. I. rnold To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected
More informationLarge 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 informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More informationOn an antiramsey type result
On an antiramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider antiramsey type results. For a given coloring of the kelement subsets of an nelement set X, where two kelement
More informationI. 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 informationSEQUENCES 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 informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationLabeling 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 informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More informationThe 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 informationClass One: Degree Sequences
Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of
More information3. 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 informationA 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 HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationDivisor 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 information136 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 informationPROBLEM 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 informationGraphs 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 informationTriangle 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 information6.852: Distributed Algorithms Fall, 2009. Class 2
.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparisonbased algorithms. Basic computation in general synchronous networks: Leader election
More informationDegree 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 informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationUniform Multicommodity Flow through the Complete Graph with Random Edgecapacities
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 informationOn 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 informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationProduct irregularity strength of certain graphs
Also available at http://amc.imfm.si ISSN 18553966 (printed edn.), ISSN 18553974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer
More informationOutline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1
GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept
More informationSimple Graphs Degrees, Isomorphism, Paths
Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week MultiGraph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics
More informationCycles 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 CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationCS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010
CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison GomoryHu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected
More informationarxiv:1202.5749v1 [cs.ds] 26 Feb 2012
Fixedparameter tractability of multicut in directed acyclic graphs Stefan Kratsch Marcin Pilipczuk Micha l Pilipczuk Magnus Wahlström arxiv:1202.5749v1 [cs.ds] 26 Feb 2012 Abstract The Multicut problem,
More informationSocial Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationResource 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 information1 Definitions. Supplementary Material for: Digraphs. Concept graphs
Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.
More information2partitiontransitive tournaments
partitiontransitive tournaments Barry Guiduli András Gyárfás Stéphan Thomassé Peter Weidl Abstract Given a tournament score sequence s 1 s s n, we prove that there exists a tournament T on vertex set
More informationA threshold for the MakerBreaker clique game
A threshold for the MakerBreaker clique game Tobias Müller Miloš Stojaković October 7, 01 Abstract We study the MakerBreaker kclique game played on the edge set of the random graph G(n, p. In this game,
More informationmost 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia
Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle
More informationAn Overview of the Degree/Diameter Problem for Directed, Undirected and Mixed Graphs
Mirka Miller (invited speaker) University of Newcastle, Australia University of West Bohemia, Pilsen King s College, London Abstract A wellknown fundamental problem in extremal graph theory is the degree/diameter
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationA 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 sublineartime algorithm for testing whether a bounded degree graph is bipartite
More informationDisjoint Compatible Geometric Matchings
Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,
More informationOdd 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 longstanding
More informationLecture 22: November 10
CS271 Randomness & Computation Fall 2011 Lecture 22: November 10 Lecturer: Alistair Sinclair Based on scribe notes by Rafael Frongillo Disclaimer: These notes have not been subjected to the usual scrutiny
More informationON 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 informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationGraph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More informationINDISTINGUISHABILITY 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 informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationDegreeassociated reconstruction parameters of complete multipartite graphs and their complements
Degreeassociated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertexdeleted subgraphofagraphgisacard.
More informationarxiv: v2 [math.co] 30 Nov 2015
PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out
More informationCycles and cliqueminors in expanders
Cycles and cliqueminors 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 informationIntroduction to Graph Theory
Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More informationStationary 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 nonnegative
More informationarxiv:1203.1525v1 [math.co] 7 Mar 2012
Constructing subset partition graphs with strong adjacency and endpoint count properties Nicolai Hähnle haehnle@math.tuberlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationOn the crossing number of K m,n
On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known
More informationR u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c
R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 22010, January 24, 2010 RUTCOR Rutgers Center for Operations
More informationOptimal Index Codes for a Class of Multicast Networks with Receiver Side Information
Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net
More informationSHORT 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 information3. 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 informationOn 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 informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationAnswers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the kcenter algorithm first to D and then for each center in D
More informationCacti with minimum, secondminimum, and thirdminimum Kirchhoff indices
MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 4758 (2010) Cacti with minimum, secondminimum, and thirdminimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang
More informationZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS
ZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationKings in Tournaments. Yu Yibo, Di Junwei, Lin Min
Kings in Tournaments by Yu Yibo, i Junwei, Lin Min ABSTRACT. Landau, a mathematical biologist, showed in 1953 that any tournament T always contains a king. A king, however, may not exist. in the. resulting
More informationEvery 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 informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS 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 informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationExtremal Wiener Index of Trees with All Degrees Odd
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287292 ISSN 03406253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of
More informationP. Jeyanthi and N. Angel Benseera
Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONEPOINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel
More informationCS2 Algorithms and Data Structures Note 11. BreadthFirst Search and Shortest Paths
CS2 Algorithms and Data Structures Note 11 BreadthFirst Search and Shortest Paths In this last lecture of the CS2 Algorithms and Data Structures thread we will consider the problem of computing distances
More informationREPRESENTATION THEORY WEEK 12
REPRESENTATION THEORY WEEK 12 1. Reflection functors Let Q be a quiver. We say a vertex i Q 0 is +admissible if all arrows containing i have i as a target. If all arrows containing i have i as a source,
More informationMinimally Infeasible Set Partitioning Problems with Balanced Constraints
Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of
More informationA NOTE ON OFFDIAGONAL SMALL ONLINE RAMSEY NUMBERS FOR PATHS
A NOTE ON OFFDIAGONAL SMALL ONLINE RAMSEY NUMBERS FOR PATHS PAWE L PRA LAT Abstract. In this note we consider the online Ramsey numbers R(P n, P m ) for paths. Using a high performance computing clusters,
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationDefinition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
More informationOn Pebbling Graphs by their Blocks
On Pebbling Graphs by their Blocks Dawn Curtis, Taylor Hines, Glenn Hurlbert, Tatiana Moyer Department of Mathematics and Statistics Arizona State University, Tempe, AZ 852871804 November 19, 2008 dawn.curtis@asu.edu
More information1. 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 MenGen 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 informationx 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