# On the Sum Necessary to Ensure that a Degree Sequence is Potentially H-Graphic

Save this PDF as:

Size: px
Start display at page:

Download "On the Sum Necessary to Ensure that a Degree Sequence is Potentially H-Graphic"

## Transcription

1 On the Sum Necessary to Ensure that a Degree Sequence is Potentially H-Graphic Michael J. Ferrara 1,2 Timothy D. LeSaulnier 3 Casey K. Moffatt 1 Paul S. Wenger 4 September 2, 2014 Abstract Given a graph H, a graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question: Determine the minimum even integer σ(h, n) such that every n-term graphic sequence with sum at least σ(h, n) is potentially H-graphic. This problem can be viewed as a potential degree sequence relaxation of the (forcible) Turán problems. While the exact value of σ(h, n) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary H. In this paper, we determine σ(h, n) asymptotically for all H, thereby providing an Erdős-Stone-Simonovits-type theorem for the Erdős-Jacobson-Lehel problem. Keywords: Degree sequence, potentially H-graphic sequence 1 Introduction A sequence of nonnegative integers π = (d 1, d 2,..., d n ) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realize or be a realization of π, and we will write π = π(g). If a sequence π consists of the terms d 1,..., d t having multiplicities µ 1,..., µ t, we may write π = (d µ 1 1,..., d µt t ). Unless otherwise noted, throughout 1 Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO ; addresses: 2 Research Supported in part by Simons Foundation Collaboration Grant # Department of Mathematics, University of Illinois, Urbana, IL 61801; 4 School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623; 1

2 this paper all sequences are nonincreasing. Additionally, we let σ(π) denote the sum of the terms of π. The study of graphic sequences dates to the 1950s, and includes the characterization of graphic sequences by Havel [19] and Hakimi [17] and an independent characterization by Erdős and Gallai [10]. Subsequent research sought to describe those graphic sequences that are realized by graphs with certain desired properties. Such problems can be broadly classified into two types, first described as forcible problems and potential problems by A.R. Rao in [29]. In a forcible degree sequence problem, a specified graph property must exist in every realization of the degree sequence π, while in a potential degree sequence problem, the desired property must be found in at least one realization of π. Results on forcible degree sequences are often stated as traditional problems in structural or extremal graph theory, where a necessary and/or sufficient condition is given in terms of the degrees of the vertices (or equivalently the number of edges) of a given graph (e.g. Dirac s Theorem on hamiltonian graphs or the number of edges in a maximal planar graph). Two older, but exceptionally thorough surveys on forcible and potential problems are due to Hakimi and Schmeichel [18] and S.B. Rao [30]. A number of degree sequence analogues to classical problems in extremal graph theory appear throughout the literature, including potentially graphic sequence variants of Hadwiger s Conjecture [7, 31], the Sauer-Spencer graph packing theorem [1], and the Erdős-Sós Conjecture [26]. Pertinent to our work here is the Turán Problem, one of the most well-established and central problems in extremal graph theory. Problem 1 (The Turán Problem). Let H be a graph and n be a positive integer. Determine the minimum integer ex(h, n) such that every graph of order n with at least ex(h, n) + 1 edges contains H as a subgraph. We refer to ex(h, n) as the extremal number or extremal function of H. Mantel [27] determined ex(k 3, n) in 1907 and Turán [32] determined ex(k t, n) for all t 3 in 1941, a result considered by many to mark the start of modern extremal graph theory. Outside of these results, the exact value of the extremal function is known for very few graphs (cf. [2, 4, 9]). In 1966, however, Erdős and Simonovits [12] extended previous work of Erdős and Stone [13] and determined ex(h, n) asymptotically for arbitrary H. More precisely, this seminal theorem gives exact asymptotics for ex(h, n) when H is a nonbipartite graph. Theorem 1 (The Erdős-Stone-Simonovits Theorem). If H is a graph with chromatic number χ(h) 2, then ( ) ( ) 1 n ex(h, n) = 1 + o(n 2 ). χ(h) 1 2 Given a family F of graphs, a graphic sequence π is potentially F-graphic if there is a realization of π that contains some F F as a subgraph. If F = {H}, we say that π is potentially H-graphic. The focus of this paper is the following problem posed by Erdős, Jacobson and Lehel in 1991 [11]. 2

3 Problem 2. Given a graph H, determine σ(h, n), the minimum even integer such that every n-term graphic sequence π with σ(π) σ(h, n) is potentially H-graphic. We will refer to σ(h, n) as the potential number or potential function of H. As σ(π) is twice the number of edges in any realization of π, Problem 2 can be viewed as a potential degree sequence relaxation of the Turán problem. In [11], Erdős, Jacobson and Lehel conjectured that σ(k t, n) = (t 2)(2n t + 1) + 2. The cases t = 3, 4 and 5 were proved separately (see respectively [11], [16] and [21], and [22]), and Li, Song and Luo [23] proved the conjecture true for t 6 and n ( t 2) + 3. In addition to these results for complete graphs, the value of σ(h, n) has been determined exactly for a number of other specific graph families, including complete bipartite graphs [5, 25], disjoint unions of cliques [14], and the class of graphs with independence number two [15] (for a number of additional examples, we refer the reader to the references of [15]). Despite this, relatively little is known in general about the potential function for arbitrary H. In this paper, we determine σ(h, n) asymptotically for all H, thereby giving a potentially H-graphic sequence analogue to the Erdős-Stone-Simonovits Theorem. 2 Constructions and Statement of Main Result We assume that H is an arbitrary graph of order k with at least one nontrivial component and furthermore that n is sufficiently large relative to k. We let (F ) denote the maximum degree of a graph F, and let F < H denote that F is an induced subgraph of H. For each i {α(h) + 1,..., k} let and consider the sequence i (H) = min{ (F ) : F < H, V (F ) = i}, π i (H, n) = ((n 1) k i, (k i + i (H) 1) n k+i ). If this sequence is not graphic, that is if n k + i and i (H) 1 are both odd, we reduce the smallest term by one. To see that this yields a graphic sequence, we make two observations. First, ( i (H) 1)-regular graphs of order n k + i i (H) exist whenever i (H) 1 and n k + i are not both odd. If n k + i and i (H) 1 are both odd, it is not difficult to show that the sequence (( i (H) 1) n k+i 1, i (H) 2) is graphic. We first show that the sequence π i (H, n) is not potentially H-graphic for each i {α(h)+ 1,..., k}, thus establishing a lower bound on σ(h, n). Proposition 1. If H is a graph of order k and n is a positive integer, then σ(h, n) σ( π i (H, n)) + 2 for all i {α(h) + 1,..., k}. Proof. Every realization G of π i (H, n) is a complete graph on k i vertices joined to an (n k + i)-vertex graph G i with maximum degree i 1. Any k-vertex subgraph of G contains at least i vertices in G i. Thus H is not a subgraph of G since every i-vertex induced subgraph of H has maximum degree at least i. 3

4 The focus of this paper is the asymptotic behavior of the potential function. As such, let σ i (H) = 2(k i) + i (H) 1, which is the leading coefficient of σ( π i (H, n)). conjectured the following. In [15], the first author and J. Schmitt Conjecture 1. Let H be a graph, and let ɛ > 0. There exists an n 0 = n 0 (ɛ, H) such that for any n > n 0, σ(h, n) max H H ( σ α(h )+1(H ) + ɛ)n. The condition that one must examine subgraphs of H is necessary. As an example, for t 3 let H be obtained by subdividing one edge of K 1,t. Since K 1,t is a subgraph of H, any sequence that is potentially H-graphic is necessarily potentially K 1,t -graphic. However, both graphs have independence number t and σ t+1 (H) < σ t+1 (K 1,t ). We show that in fact one needs only examine somewhat large induced subgraphs of H. The following, which determines the asymptotics of the potential function precisely for arbitrary H, is the main result of this paper. Theorem 2. Let H be a graph of order k and let n be a positive integer. If σ(h) is the maximum of σ i (H) for i {α(h) + 1,..., k}, then σ(h, n) = σ(h)n + o(n). As was pointed out in [15], Conjecture 1 is correct for all graphs H for which σ(h, n) is known. Consequently, it is feasible that Theorem 2 is actually an affirmation of Conjecture 1. That is, for all H is it possible that σ(h) = max H H ( σ α(h )+1(H )), however we are unable to either verify or disprove this at this time. The proof of Theorem 2 is an immediate consequence of Proposition 1 and the following result. Theorem 3. Let H be a graph, and let ω = ω(n) be an increasing function that tends to infinity with n. There exists an N = N(ω, H) such that for any n N, σ(h, n) σ(h)n + ω(n). The proof of Theorem 3 relies on repeated use of the following theorem, which may be of independent interest. Let H be a graph, and let (h 1,..., h k ) be the degree sequence of H. A graphic sequence π = (d 1,..., d n ) is degree sufficient for H if d i h i for all i {1,..., k}. Theorem 4 (The Bounded Maximum Degree Theorem). Let H be a graph of order k. There exists a function f = f(α(h), k) such that, for sufficiently large n, a nonincreasing graphic sequence π = (d 1,..., d n ) is potentially H-graphic provided that 4

5 1. π is degree sufficient for H, 2. d n k α(h), and 3. d 1 < n f(α(h), k). In Section 3 we present several technical lemmas used in the proofs of Theorems 3 and 4. In Section 4 we prove the Bounded Maximum Degree Theorem, with the proof of Theorems 1 and 3 following in Section 5. 3 Technical Lemmas We will need the following results from [24] and [33]. Theorem 5 (Yin and Li). Let π = (d 1,..., d n ) be a nonincreasing graphic sequence and let k be a positive integer. If d k k 1 and d i 2(k 1) i for all i {1,..., k 1}, then π is potentially K k -graphic. G. We let G H denote the standard join of G and H and let G denote the complement of Lemma 1 (Yin). If π is a potentially K r K s -graphic sequence, then there is a realization of π in which the vertices in the copy of K r are the r vertices of highest degree and the vertices in the copy of K s are the next s vertices of highest degree. We now present a classical result of Kleitman and Wang [20] that generalizes the Havel- Hakimi algorithm [17, 19]. Theorem 6 (Kleitman and Wang). Let π = (d 1,..., d n ) be a nonincreasing sequence of nonnegative integers. If π (i) is the sequence defined by { (d 1 1,..., d di 1, d di +1,..., d i 1, d i+1,..., d n ) if d i < i π (i) = (d 1 1,..., d i 1 1, d i+1 1,..., d di +1 1, d di +2,..., d n ) if d i i, then π is graphic if and only if π (i) is graphic. The process of removing a term from a graphic sequence as described in the Kleitman- Wang algorithm is referred to as laying off the term d i from π, and the sequence π i obtained is frequently called a residual sequence. The next lemma, the proof of which is essentially identical to that of the necessity of the Havel-Hakimi and Kleitman-Wang algorithms, allows us to use the structure of a realization of π to choose the term that we wish to lay off in the Kleitman-Wang algorithm. Lemma 2. Let G be a graph with degree sequence π and let v V (G). If π(g v) is potentially H-graphic, then π is potentially H-graphic. In particular, if π i is any residual sequence obtained from π via the Kleitman-Wang algorithm and π i is potentially H-graphic, then π is potentially H-graphic. 5

6 Proof. Let π = (d 1,..., d n ) and let V (G) = {v 1,..., v n } such that d(v i ) = d i. Assume that π i is potentially H-graphic and let G be a realization of π(g v i ) that contains H as a subgraph. If d i < i, obtain a realization of π containing H by joining v i to vertices with degrees d 1 1,..., d di 1 in G. If d i i, obtain a realization of π containing H by joining v i to vertices with degrees d 1 1,..., d i 1 1, d i+1 1,..., d di +1 1 in G. In the same spirit, we also give the following useful lemma. As the proof is straightforward, we omit it here. For a graph G and an integer i with i {0,..., V (G) }, let D (i) (G) denote the family of graphs obtained by deleting exactly i vertices from G or, in other words, the family of induced subgraphs of G with order V (G) i. Lemma 3. If π = (n 1, d 2,..., d n ) is a nonincreasing graphic sequence, then π is potentially H-graphic if and only if π 1 = (d 2 1,..., d n 1) is potentially H -graphic for some H in D (1). Iteratively applying the Kleitman-Wang algorithm yields the following, which generalizes related lemmas from [3, 14]. Lemma 4. Let m and k be positive integers and let ω = ω(n) be an increasing function that tends to infinity with n. There exists N = N(m, k, ω) such that for all n N, if π is an n-term graphic sequence such that σ(π) mn + ω(n), then 1. π is potentially K k -graphic, or 2. iteratively applying the Kleitman-Wang algorithm by laying off the minimum term in each successive sequence will eventually yield a graphic sequence π with n terms satisfying the following properties: (a) σ(π ) mn + ω(n ), (b) the minimum term in π is greater than m/2, and (c) n ω(n) 2(k 2) m. Proof. First observe that if k 2, then the result is trivial, so we assume that k 3. Let π = π 0. If the minimum term of π i is greater than m/2, let π = π i. Otherwise, obtain the n i 1-term graphic sequence π i+1 by applying the Kleitman-Wang algorithm to lay off a minimum term from π i and then sorting the resulting sequence in nonincreasing order. Note that always σ(π i+1 ) σ(π i ) m, and by induction we have σ(π i+1 ) m(n i 1) + ω(n). By Lemma 2, if π i is potentially K k -graphic, then π 0 is potentially K k -graphic. By the results of [11], [16] and [21], [22], and [23], we know that σ(k k, n) = 2(k 2)n (k 1)(k 2)+2 for n ( k 2) + 3. Thus, if σ(πi ) 2(k 2)(n i) + 2 for some i 0, then π is potentially K k -graphic. Considering the case when i = 0, we may assume that m < 2(k 2). If i n ω(n), then 2(k 2) m σ(π i ) 2(k 2)(n i) (m 2(k 2))(n i) + ω(n) ( ) ω(n) (m 2(k 2)) + ω(n) 2(k 2) m 0. 6

7 ω(n) ω(n) Therefore, if i n and ( k 2(k 2) m 2(k 2) m 2) + 3, then πi is potentially K k -graphic and consequently π is potentially K k -graphic. It now follows that if π is not potentially K k -graphic, then the process of applying the Kleitman-Wang algorithm must output π = π i for some i < n ω(n). Since ω is 2(k 2) m increasing and n n, it follows that σ(π ) mn + ω(n ). As demonstrated in Lemma 4, repeatedly laying off minimum terms from a sequence via the Kleitman-Wang algorithm can allow us to obtain a residual sequence that is denser than π and also has a large minimum degree, which may facilitate the construction of a realization that contains H. At times, we may instead wish to delete vertices from specific realizations of π. Finally, for completeness, we state the classic characterization of graphic sequences due to Erdős and Gallai [10], which will be useful in the proof of Theorem 3. Theorem 7 (The Erdős-Gallai Graphicality Criteria). If π = (d 1,..., d n ) is a nonincreasing sequence of nonnegative integers, then π is graphic if and only if π has even sum and for all t {1,..., n 1}. t d j t(t 1) + j=1 n j=t+1 min{d j, t} 4 Proof of the Bounded Maximum Degree Theorem In this section, we prove the Bounded Maximum Degree Theorem (Theorem 4). Both here and in the proof of Theorem 3, we demonstrate the existence of a realization of π that either contains H or some supergraph of H, for instance K k or K k α(h) K α(h). Let e 1 = u 1 v 1 and e 2 = u 2 v 2 be edges in a graph G such that u 1 u 2 and v 1 v 2 are not in E(G). Removing e 1 and e 2 from G and replacing them with the edges u 1 u 2 and v 1 v 2 results in a graph with the same degree sequence as G. This operation is called a 2-switch, but throughout the literature on degree sequences has also been referred to as a swap, rewiring or infusion. In [28], Petersen showed that, given realizations G 1 and G 2 of a graphic sequence π, G 1 can be obtained from G 2 via a sequence of 2-switches. Both here and in Section 5, we will utilize both 2-switches and more general edge-exchange operations. Let C be a list of vertices C = v 0, v 1,..., v t with v t = v 0 in a graph G such that v i 1 v i E(G) if and only if v i v i+1 / E(G). Observe that removing the edges in E(C) E(G) from G and adding the edges in E(C) E(G) to G preserves the degree sequence of G. Proof. Let V (H) = {u 1,..., u k }, indexed such that d H (u i ) d H (u j ) when i j. Let us assume that π satisfies the hypothesis of the theorem, but is not potentially H-graphic. In a realization G of π, let S = {v 1,..., v k } be a set of vertices such that d G (v i ) = d i and let H S be the graph with vertex set S in which two vertices v i, v j are adjacent if and only u i u j E(H). If all edges of H S are edges of G, then H S is a subgraph of G isomorphic 7

8 to H and π is potentially H-graphic. Hence, let us assume that G is a realization which maximizes E(G) E(H S ), but that the edge v i v j E(H S ) while v i v j E(G). Now, because d G (v i ) d H (u i ) = d HS (v i ) and d G (v j ) d H (u j ) = d HS (v j ), there must exist (not necessarily distinct) vertices a i and a j such that v i a i, v j a j E(G), while v i a i, v j a j E(H S ). We begin by showing that many vertices of V (G) S have neighbors in V (G) S. Specifically, we claim that there are at most ( ) [ ( ) k k α(h) g(α(h), k) = 2 k α(h) 2 ] + α(h) 1 vertices w in V (G) S such that N(w) S. Indeed, let us assume that there are at least g(α(h), k) + 1 vertices w in V (G) S such that N(w) S. By the hypothesis of the theorem, each vertex w V (G) satisfies d G (w) k α(h). For each vertex w such that N(w) S, let S w be a set of k α(h) vertices such that S w N(w). Let l = 2 ( ) k α(h) 2 + α(h). By the pigeonhole principle, there exists an independent set {w 1,..., w l } of vertices and a k α(h) element subset Ŝ of S such that S wi = Ŝ for each 1 i l. Let P be a family of ( ) k α(h) 2 disjoint pairs of vertices from {w1,..., w l α(h) }, and to each pair {v r, v s } of vertices in Ŝ associate a distinct pair P (r, s) P. If v rv s is not an edge in G and we suppose P (r, s) = {w r.w s }, then we perform the 2-switch that replaces the edges v r w r and v s w s with the non-edges v r v s and w r w s. In this way arrive at a realization of π in which the graph induced by Ŝ is complete, while maintaining the property that the vertices of Ŝ are joined to {w l α(h)+1...., w l }. This produces a realization of π that contains K k α(h) K α(h) as a subgraph, contradicting the assumption that π is not potentially H- graphic. Consequently, we may assume that there are at most g(α(h), k) vertices w in V (G) S such that N(w) S. We will now use the fact that many vertices of V (G) S have neighbors in V (G) S to exhibit an edge-exchange that inserts the edge v i v j into G at the expense of the edges v i a i and v j a j while preserving each edge in H S. Let and let d 1 n 1 f(α(h), k). If we let and f(α(h), k) = [g(α(h), k) + 4k 2 ] + k + 1, X i = {v (V (G) S) N G S (a i ) : d G S (v) > 0} X j = {v (V (G) S) N G S (a j ) : d G S (v) > 0}, then, as at most g(α, k) vertices have their neighborhoods entirely contained in S, it follows that X i, X j 4k 2. Let Y i = N G S (X i ) and Y j = N G S (X j ). 8

9 By assumption, π is not potentially H-graphic, and is therefore not potentially K k - graphic. As such, Theorem 5 implies that each vertex in G S has degree at most 2k 4 < 2k. Let y i be a vertex in Y i, and let x i be a neighbor of y i in X i. There are at least 4k 2 1 vertices in X j that are not x i, and at least 4k 2 1 (2k 5) of these vertices have a neighbor in Y j that is not y i. Since vertices in V (G) S have degree at most 2k 4, there are more than (4k 2 2k + 4)/(2k) = 2k 1 + 2/k vertices in Y j that are not y i and furthermore have a neighbor in X j that is not x i. Since d(y i ) < 2k, there exists a vertex y j, distinct from y i such that y i y j / E(G) and y j has a neighbor x j X j that is distinct from x i. In this case, exchanging the edges v i a i, v j a j, x i y i and x j y j for the nonedges v i v j, a i x i, a j x j and y i y j yields a realization of π that contradicts the maximality of G. 5 Proof of Theorem 3 The proof of Theorem 3 proceeds in two stages, which we briefly outline here to give a clearer picture of the structure and philosophy of the proof. Let n be sufficiently large, and let π be an n-term graphic sequence such that σ(π) σ(h)n + ω(n). Most simply stated, we prove that π has a realization in which a set of l vertices is completely joined to a large subgraph whose degree sequence is potentially D (l) (H)-graphic, for some l {0,..., k α(h)}. The first stage of the proof generates a residual sequence π l through l alternating applications of Lemma 4 and (the contrapositive of) Theorem 4. The ith application of Lemma 4 raises the average of the terms in the sequence, making the sequence denser and therefore more likely to be potentially D (i) (H)-graphic. We call this sequence π i. If π i is not potentially D (i) (H)-graphic, then the contrapositive of Theorem 4 implies that the leading term of π i is very large. Hence, in any realization G of π i there is a vertex v that is adjacent to nearly all other vertices in G. While the leading term of π i prevents the direct application of Theorem 4 to build an element of D (i) (H), in principle the corresponding vertex of high degree should be beneficial to the construction of an element of D (i) (H). We then lay off the leading term of π i, and given its utility, consider it to be reserved for the conclusion of the proof. After appropriate conditions are met, Stage 1 terminates with π l. Stage 2 of the proof then uses edge exchanges to prove that π l is potentially D (l) (H)-graphic. Since Stage 1 was iterated l times, we have laid off and reserved l large terms. Reuniting these l terms with a realization of π l that contains a subgraph from D (l) (H) allows us to create a realization of π that contains H as a subgraph, completing the proof. We are now ready to give the proof of Theorem 3 in full detail. Proof. To begin, let π be an n-term graphic sequence with n > N(ω, H) such that σ(π) σ(h)n + ω(n). It suffices to show that π is potentially H-graphic. If π is potentially K k - graphic, then the result is trivial. Therefore, by Theorem 5, we may assume that d k+1 2k 4. Furthermore, throughout the proof we may assume that when applying Lemma 4 to π or a residual sequence obtained from π, the resulting sequence satisfies Conclusion (2) of that lemma. 9

10 To initialize Stage 1, apply Lemma 4 to π to obtain an n 0 -term graphic sequence π 0 = (d (0) 1,..., d (0) n 0 ) with minimum entry at least σ(h)/2 and σ(π 0 ) σ(h)n 0 + ω(n 0 ). Starting with this π 0, we iteratively construct sequences π i = (d (i) 1,..., d (i) n i ) for i 1. Each π i, which we assume to be nonincreasing, will satisfy the following conditions: (a) d (i) n i k i α(h), and (b) if π i is potentially D (i) (H)-graphic, then π is potentially H-graphic. Note that Condition (a) ensures that π i satisfies Condition 2 of Theorem 4 for any (k i)- vertex graph H i with α(h i ) α(h). If i = k α(h) or if d (i) 1 < n i ( k k/2 ) (k 2 1), then terminate Stage 1 and proceed to Stage 2 with l = i. Otherwise, d (i) 1 n i ( k k/2 ) (k 2 1), and we obtain π i+1 via the following steps; 1. Let Ĝi be a realization of π i such that a vertex of degree d (i) 1, which we call v (i) 1, is adjacent to the next d (i) 1 vertices of highest degree; such a realization exists following the proof of the sufficiency of Theorem 6. From Ĝi, obtain the graph Ĝ i by deleting the nonneighbors of v (i) 1 and let π i = π(ĝ i). If we let n i denote the number of terms in π i, then the largest term in π i is n i Apply Lemma 4 to π i, and let π i be the graphic sequence from Conclusion (2) of that be the number of i 1. lemma. The minimum term of π i is at least k i α(h). Let n i terms in π i ; the largest term in π i will be n 3. Obtain π i+1 by laying off the largest term in π i. Note that even though Lemma 4 may require laying off a large fraction of the terms in π i, we still have that n i+1 provided n i. Thus we may assume that n i+1 is sufficiently large. We claim that σ(π i ) ( σ(h) 2i)n i + ω(n i )/2 i and the minimum term in π i is at least σ(h)/2 i k i α(h) for all i {0..., l}. We use induction on i. First observe that σ(π 0 ) σ(h)n 0 + ω(n 0 ) and (as we assume that Conclusion (2) results from all applications of Lemma 4) the minimum term of π 0 is at least σ(h)/2 k α(h). Let ( ) k M = 2 (k 2 1)(2k 4). k/2 Creating π i entails deleting at most ( k k/2 ) (k 2 1) vertices each with degree at most 2k 4, and consequently, by induction, σ(π i) σ(π i ) M ( σ(h) 2i)n i + ω(n i )/2 i M. 10

11 By assumption, n i is sufficiently large, so we have ω(n i )/2 i 2M, and therefore σ(π i) ( σ(h) 2i)n i + ω(n i )/2 i+1. We next apply Lemma 4 to π 1 to obtain a sequence π i with n i terms. It follows that σ(π i ) ( σ(h) 2i)n i + ω(n i )/2 i+1 and we also have that the minimum term in π i is at least k i α(h). The maximum term in π i is n i 1, and we lay off this term to obtain π i+1. This yields σ(π i+1 ) ( σ(h) 2i)n i + ω(n i )/2 i+1 2(n i 1) ( σ(h) 2(i + 1))n i+1 + ω(n i+1 )/2 i+1. Also, as the minimum term in π i is at least k i α(h) and decreases by 1 when the maximum term is laid off, we have that π i+1 satisfies Condition (a). Next we show that Condition (b) holds, also via induction on i. If we let D (0) (H) = {H}, then the claim holds for i = 0. For some i < l, assume that if π i is potentially D (i) (H)- graphic, then π is potentially H-graphic. By Lemma 3, π i+1 is potentially D (i+1) (H)-graphic if and only if π i is potentially D (i) (H)-graphic. By Lemma 2, if π i is potentially D (i) (H)- graphic, then π i is potentially D (i) (H)-graphic. Therefore, by induction, if π i+1 is potentially D (i+1) (H)-graphic, then π is potentially H-graphic, so π i+1 satisfies Condition (b). We now begin Stage 2 of the proof: it remains to show that π l is potentially D (l) (H)- graphic. If l = k α(h), then D (l) (H) contains K α(h), and since π l is sufficiently long the ( result holds trivially. Thus we may assume that l < k α(h) and that d (l) 1 < n l k k/2 ) (k 2 1). In π l, let t = max{i : d (l) i k l 1}. First assume that t k l α(h). In this case, π l is degree sufficient for K k l α(h) K α(h). Furthermore, the minimum term in π l is at least k l α(h), and d (l) 1 < n l ( k k/2 ) (k 2 1) < n l f(α(h), k l). Thus Theorem 4 implies that π l is potentially K k l α(h) K α(h) -graphic. Since K k l α(h) K α(h) is a supergraph of a (k l)-vertex subgraph of H, it follows that π l is potentially D (l) (H)-graphic. Now assume that t < k l α(h). Let F i denote an i-vertex induced subgraph of H that achieves (F i ) = i (H). We show that π l is degree sufficient for K t F k l t. First observe that since k l t > α(h) we have σ(h) 2l σ k l t (H) 2l = 2t k l t (H) 1. Thus, σ(π l ) (2t k l t (H) 1)n l + ω(n l )/2 l. However, if π l is not degree sufficient for K t F k l t, then since the minimum degree of K t F k l t is at most t + k l t (H), it follows that d (l) k l t + k l t(h) 1. Taken together with the fact that d (l) t+1 k l 2, we then have that σ(π l ) t(n l 1) + (k l t 1)(k l 2) + (n l k + l + 1)(t + k l t (H) 1). As l k 1, this implies that σ(π l ) < (2t + k l t (H) 1)n l + k 2, 11

12 contradicting the above lower bound when n l is sufficiently large. Consequently, we have that π l is degree sufficient for K t F k l t. Note that α(f k l t ) may be less than α(h). Therefore we are not able to immediately apply Theorem 4 to obtain a realization of π l containing K t F k l t. However, in this case, π l is degree sufficient for K t K k l t. Since π l satisfies Condition (a), the minimum term of π l is at least k l α(h), and by assumption, k l α(h) > t. Furthermore, d (l) 1 < n l ( k k/2 ) (k 2 1) < n l f(k l t, k l). Thus by Theorem 4, there is a realization G l of π l containing K t K k l t. If v 1,..., v k l are the k l vertices of highest degree in G l, then by Lemma 1 we may assume that {v 1,..., v t } is a clique that is completely joined to {v t+1,..., v k l }. Delete v 1,..., v t from G l to obtain G l, and let π l = π(g l ), with the order of π l coming from the ordering of π l. Thus the first k l t terms in π l correspond to the vertices {v t+1,..., v k l } in G l. Since the minimum degree of the vertices in G l is at least k l α(h) and t < k l α(h), the minimum term in π l is at least 1. It remains to show that there is a realization of π l that contains a copy of F k l t on the vertices {v t+1,..., v k l }. To construct such a realization, place a copy of F k l t on the vertices v t+1,..., v k l. Since n l is sufficiently large, we may join any remaining edges incident to {v t+1,..., v k l } to distinct vertices among the remaining n l (k l) vertices. It remains to show that there is a graph on the remaining n l (k l) vertices that realizes the residual sequence. This sequence has at least n l (k l) (k l t)(k l 3) positive terms with the maximum term being at most k l 2. By Theorem 7, the Erdős-Gallai criteria, such a sequence is graphic provided that n l is sufficiently large. As noted above, Theorem 2 follows immediately from Theorems 1 and 3. 6 Conclusion Having determined the asymptotic value of the potential function for general H, a natural next step is to study the structure of those n-term graphic sequences that are not potentially H-graphic, but whose sum is close to σ(h, n). This line of inquiry would be related to recent work of Chudnovsky and Seymour [6], which for an arbitrary graphic sequence π gives a partial structural characterization of those graphic sequences π for which no realization of π contains any realization of π as an induced subgraph. The first stage of this investigation appears in [8]. Acknowledgement: The authors would like to acknowledge the detailed and extremely helpful efforts of an anonymous referee whose suggestions have greatly improved the clarity and presentation of this paper. 12

13 References [1] A. Busch, M. Ferrara, M. Jacobson, H. Kaul, S. Hartke and D. West, Packing of Graphic n-tuples, J. Graph Theory 70 (2012), [2] N. Bushaw and N. Kettle. Turán numbers of Multiple Paths and Equibipartite Trees, Combin. Probab. Comput. 20 (2011), [3] G. Chen, M. Ferrara, R. Gould and J. Schmitt, Graphic Sequences with a Realization Containing a Complete Multipartite Subgraph, Discrete Math. 308 (2008), [4] G. Chen, R. Gould, F. Pfender and B. Wei, Extremal Graphs for Intersecting Cliques, J. Combin. Theory Ser. B 89 (2003), [5] G. Chen, J. Li and J. Yin, A variation of a classical Turán-type extremal problem, European J. Comb. 25 (2004), [6] M. Chudnovsky and P. Seymour, Rao s Degree Sequence Conjecture, J. Combin. Theory Ser. B 105 (2014), [7] Z. Dvořák and B. Mohar, Chromatic number and complete graph substructures for degree sequences, Combinatorica 33 (2013) [8] C. Erbes, M. Ferrara, R. Martin and P. Wenger, On the Approximate Shape of Degree Sequences the are not Potentially H-graphic, submitted. arxiv: [9] P. Erdős, Z. Füredi, R. Gould and D. Gunderson, Extremal Graphs for Intersecting Triangles, J. Combin. Theory Ser. B 64 (1995), [10] P. Erdős and T. Gallai, Graphs with prescribed degrees, Matematiki Lapor 11 (1960), (in Hungarian). [11] P. Erdős, M. Jacobson and J. Lehel, Graphs Realizing the Same Degree Sequence and their Respective Clique Numbers, Graph Theory, Combinatorics and Applications (eds. Alavi, Chartrand, Oellerman and Schwenk), Vol. I, 1991, [12] P. Erdős and M. Simonovits, A Limit Theorem in Graph Theory, Studia Sci. Math. Hungar. 1 (1966), [13] P. Erdős and A. Stone, On the Structure of Linear Graphs, Bull. Amer. Math. Soc. 52 (1946), [14] M. Ferrara, Graphic Sequences with a Realization Containing a Union of Cliques, Graphs Comb. 23 (2007), [15] M. Ferrara and J. Schmitt, A General Lower Bound for Potentially H-Graphic Sequences, SIAM J. Discrete Math. 23 (2009),

14 [16] R. Gould, M. Jacobson, and J. Lehel, Potentially G-graphic degree sequences, Combinatorics, Graph Theory, and Algorithms (eds. Alavi, Lick and Schwenk), Vol. I, New York: Wiley & Sons, Inc., 1999, [17] S.L. Hakimi, On the realizability of a set of integers as degrees of vertices of a graph, J. SIAM Appl. Math, 10 (1962), [18] S. Hakimi and E. Schmeichel, Graphs and their degree sequences: A survey. In Theory and Applications of Graphs, volume 642 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1978, [19] V. Havel, A remark on the existence of finite graphs (Czech.), Časopis P ěst. Mat. 80 (1955), [20] D. Kleitman and D. Wang, Algorithms for constructing graphs and digraphs with given valences and factors, Discrete Math. 6 (1973) [21] J. Li and Z. Song, An extremal problem on the potentially P k -graphic sequences, The International Symposium on Combinatorics and Applications (W.Y.C. Chen et. al., eds.), Tanjin, Nankai University 1996, [22] J. Li and Z. Song, The smallest degree sum that yields potentially P k -graphical sequences, J. Graph Theory 29 (1998), [23] J. Li, Z. Song, and R. Luo, The Erdős-Jacobson-Lehel conjecture on potentially P k - graphic sequences is true, Science in China, Ser. A, 41 (1998), [24] J. Li and J. Yin, Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size, Discrete Math. 301 (2005), [25] J. Li and J. Yin, An extremal problem on potentially K r,s -graphic sequences, Discrete Math. 260 (2003), [26] J. Li and J. Yin, A variation of a conjecture due to Erdős and Sós, Acta Math. Sin. 25 (2009), [27] W. Mantel, Problem 28, soln by H. Gouwentak, W. Mantel, J Teixeira de Mattes, F. Schuh and W.A. Wythoff, Wiskundige Opgaven 10 (1907), [28] J. Peteresen, Die Theorie der regularen Graphen, Acta Math. 15 (1891), [29] A. R. Rao, The clique number of a graph with a given degree sequence. In Proceedings of the Symposium on Graph Theory, volume 4 of ISI Lecture Notes, Macmillan of India, New Delhi, 1979, [30] S. B. Rao, A survey of the theory of potentially P -graphic and forcibly P -graphic degree sequences. In Combinatorics and graph theory, volume 885 of Lecture Notes in Math., Springer, Berlin, 1981,

15 [31] N. Robertson and Z. Song, Hadwiger number and chromatic number for near regular degree sequences, J. Graph Theory 64 (2010), [32] P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapook 48 (1941), [33] J. Yin, A Rao-type characterization for a sequence to have a realization containing a split graph, Discrete Math. 311 (2011),

### RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES

RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES ARTHUR BUSCH UNIVERSITY OF DAYTON DAYTON, OH 45469 MICHAEL FERRARA, MICHAEL JACOBSON 1 UNIVERSITY OF COLORADO DENVER DENVER, CO 80217 STEPHEN G. HARTKE 2 UNIVERSITY

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### Best 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

### 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

### 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.,

### A maximum degree theorem for diameter-2-critical graphs

Cent. Eur. J. Math. 1(1) 01 188-1889 DOI: 10.78/s11533-01-09-3 Central European Journal of Mathematics A maximum degree theorem for diameter--critical graphs Research Article Teresa W. Haynes 1,, Michael

### 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:

### 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

### 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

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### 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

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### 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

### Some Problems on Graphic Sequences

Some Problems on Graphic Sequences Michael Ferrara 1 Department of Mathematical and Statistical Sciences University of Colorado Denver Denver, CO 80217 michael.ferrara@ucdenver.edu Dedicated to Charles

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

### 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

### 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

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### Generalizing the Ramsey Problem through Diameter

Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive

### Coloring Eulerian triangulations of the projective plane

Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@uni-lj.si Abstract A simple characterization

### 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

### Intersection Dimension and Maximum Degree

Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai - 600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show

### Rational exponents in extremal graph theory

Rational exponents in extremal graph theory Boris Bukh David Conlon Abstract Given a family of graphs H, the extremal number ex(n, H) is the largest m for which there exists a graph with n vertices and

### 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

### 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

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Degree sequences forcing Hamilton cycles in directed graphs

Degree sequences forcing Hamilton cycles in directed graphs Daniela Kühn 1,2 School of Mathematics University of Birmingham Birmingham, UK Deryk Osthus & Andrew Treglown 1,3 School of Mathematics University

### 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

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### Removing Even Crossings

EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### 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].

### 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

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### Definition 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

### Finding 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

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### Extremal Graphs without Three-Cycles or Four-Cycles

Extremal Graphs without Three-Cycles or Four-Cycles David K. Garnick Department of Computer Science Bowdoin College Brunswick, Maine 04011 Y. H. Harris Kwong Department of Mathematics and Computer Science

### PRIME FACTORS OF CONSECUTIVE INTEGERS

PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

### An inequality for the group chromatic number of a graph

Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

### 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

### A Study of Sufficient Conditions for Hamiltonian Cycles

DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph

### 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

### 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

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES JONATHAN CHAPPELON, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract Let K [k,t] be the complete graph on k vertices from which a set of edges,

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### M-Degrees of Quadrangle-Free Planar Graphs

M-Degrees of Quadrangle-Free Planar Graphs Oleg V. Borodin, 1 Alexandr V. Kostochka, 1,2 Naeem N. Sheikh, 2 and Gexin Yu 3 1 SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK 630090, RUSSIA E-mail: brdnoleg@math.nsc.ru

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs

Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs C. Boucher D. Loker May 2006 Abstract A problem is said to be GI-complete if it is provably as hard as graph isomorphism;

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

### 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

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### 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

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### 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

### Available online at ScienceDirect. Procedia Computer Science 74 (2015 ) 47 52

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and

### 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

### A simple existence criterion for normal spanning trees in infinite graphs

1 A simple existence criterion for normal spanning trees in infinite graphs Reinhard Diestel Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the

### ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

### Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Degree-associated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertex-deleted subgraphofagraphgisacard.

### 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

### 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

### The Independence Number in Graphs of Maximum Degree Three

The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684

### Set systems with union and intersection constraints

Set systems with union and intersection constraints Dhruv Mubayi Reshma Ramadurai October 15, 2008 Abstract Let 2 d k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets

### 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

### Online Degree Ramsey Theory

UofL Discrete Math Workshop 2008 1 Online Degree Ramsey Theory posed by Illinois REGS (2007) problem 1 presented by Lesley Wiglesworth LATEX byadamjobson For a family of graphs F closed under subgraphs,

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### On a tree graph dened by a set of cycles

Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor Neumann-Lara b, Eduardo Rivera-Campo c;1 a Center for Combinatorics,

### 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

### Some Results on 2-Lifts of Graphs

Some Results on -Lifts of Graphs Carsten Peterson Advised by: Anup Rao August 8, 014 1 Ramanujan Graphs Let G be a d-regular graph. Every d-regular graph has d as an eigenvalue (with multiplicity equal

### Diameter of paired domination edge-critical graphs

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 279 291 Diameter of paired domination edge-critical graphs M. Edwards R.G. Gibson Department of Mathematics and Statistics University of Victoria

### Zero-Sum Magic Labelings and Null Sets of Regular Graphs

Zero-Sum Magic Labelings and Null Sets of Regular Graphs Saieed Akbari a,c Farhad Rahmati b Sanaz Zare b,c a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department

### Multiply intersecting families of sets

Multiply intersecting families of sets Zoltán Füredi 1 Department of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL61801, USA and Rényi Institute of Mathematics of the Hungarian Academy

### Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

### 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

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### 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

### 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

### princeton 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

### 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

### 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

### 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

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### 6.2 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

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### 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