On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Conditions on Ramsey Nonequivalence

Given a graph H , a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G , H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. (J Combin Theory Ser B 109 (2014), 120–133) asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Ne?et?il et al. showed that any two graphs with different clique number (Combinatorica 1(2) (1981), 199–202) or different odd girth (Comment Math Univ Carolin 20(3) (1979), 565–582) are not Ramsey equivalent. These are the only structural graph parameters we know that “distinguish” two graphs in the above sense. This article provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, the chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most five vertices are not Ramsey equivalent to any other connected graph. Moreover, two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique‐reduction properties.     

On an anti‐Ramsey problem of Burr,Erdős,Graham, and T. Sós

Given a graph L, in this article we investigate the anti‐Ramsey number χ_S(n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G(n,e) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interesting results and problems, Burr, Erd?s, Graham, and T. Sós asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that In this article, we prove a slightly weaker statement, namely we show that the statement is true if L is a connected bipartite graph, which is not a complete bipartite graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 147–156, 2006     

Linear Ramsey numbers of sparse graphs

The Ramsey number R(G₁,G₂) of two graphs G₁ and G₂ is the least integer p so that either a graph G of order p contains a copy of G₁ or its complement G^c contains a copy of G₂. In 1973, Burr and Erd?s offered a total of $25 for settling the conjecture that there is a constant c = c(d) so that R(G,G)≤ c|V(G)| for all d‐degenerate graphs G, i.e., the Ramsey numbers grow linearly for d‐degenerate graphs. We show in this paper that the Ramsey numbers grow linearly for degenerate graphs versus some sparser graphs, arrangeable graphs, and crowns for example. This implies that the Ramsey numbers grow linearly for degenerate graphs versus graphs with bounded maximum degree, planar graphs, or graphs without containing any topological minor of a fixed clique, etc. © 2005 Wiley Periodicals, Inc. J Graph Theory     

There exist graphs with super‐exponential Ramsey multiplicity constant

The Ramsey multiplicity M(G;n) of a graph G is the minimum number of monochromatic copies of G over all 2‐colorings of the edges of the complete graph K_n. For a graph G with a automorphisms, ν vertices, and E edges, it is natural to define the Ramsey multiplicity constant C(G) to be , which is the limit of the fraction of the total number of copies of G which must be monochromatic in a 2‐coloring of the edges of K_n. In 1980, Burr and Rosta showed that 0 ≥ C(G) ≤ 2^1?E for all graphs G, and conjectured that this upper bound is tight. Counterexamples of Burr and Rosta's conjecture were first found by Sidorenko and Thomason independently. Later, Clark proved that there are graphs G with E edges and 2^E?1C(G) arbitrarily small. We prove that for each positive integer E, there is a graph G with E edges and C(G) ≤ E^{?E/2 + o(E)}. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 89–98, 2008     

On chromatic‐choosable graphs

A graph is chromatic‐choosable if its choice number coincides with its chromatic number. It is shown in this article that, for any graph G, if we join a sufficiently large complete graph to G, then we obtain a chromatic‐choosable graph. As a consequence, if the chromatic number of a graph G is close enough to the number of vertices in G, then G is chromatic‐choosable. We also propose a conjecture related to this fact. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 130–135, 2002     

Multipartite Ramsey numbers for odd cycles

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic C_n, a cycle of length n. This roughly says that the Ramsey number for C_n (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K_2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009     

Constrained Ramsey numbers for rainbow matching

The Ramsey number R_k(G) of a graph G is the minimum number N, such that any edge coloring of K_N with k colors contains a monochromatic copy of G. The constrained Ramsey number f(G, T) of the graphs G and T is the minimum number N, such that any edge coloring of K_N with any number of colors contains a monochromatic copy of G or a rainbow copy of T. We show that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK₂) = R_{t ? 1}(G) for t≥2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:91‐95, 2011     

Induced subgraphs of prescribed size

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erd?s and McKay we show that every graph G on n vertices for which q(G)≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n^{δ (C)}. Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and . © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 239–251, 2003     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

A conjecture of Erd?s on graph Ramsey numbers

The Ramsey number r(G) of a graph G is the minimum N such that every red–blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics.One of the oldest results in Ramsey Theory, proved by Erd?s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most . Motivated by this estimate Erd?s conjectured, more than a quarter century ago, that there is an absolute constant c such that for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.     

Ramsey unsaturated and saturated graphs

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles C_n and paths P_n on n vertices are Ramsey unsaturated for all n ≥ 5. © 2005 Wiley Periodicals, Inc.     

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004     

Tree decompositions of graphs without large bipartite holes

A recent result of Condon, Kim, Kühn, and Osthus implies that for any , an n‐vertex almost r‐regular graph G has an approximate decomposition into any collections of n‐vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost αn‐regular graph G with any α>0 and a collection of bounded degree trees on at most (1?o(1))n vertices if G does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into n‐vertex trees. Moreover, this implies that for any α>0 and an n‐vertex almost αn‐regular graph G, with high probability, the randomly perturbed graph has an approximate decomposition into all collections of bounded degree trees of size at most (1?o(1))n simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey‐Turán theory and the randomly perturbed graph model.     

Ramsey‐type results for Gallai colorings

A Gallai‐coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai‐colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai‐colorings extend 2‐colorings of the edges of complete graphs. They actually turn out to be close to 2‐colorings—without being trivial extensions. Here, we give a method to extend some results on 2‐colorings to Gallai‐colorings, among them known and new, easy and difficult results. The method works for Gallai‐extendible families that include, for example, double stars and graphs of diameter at most d for 2?d, or complete bipartite graphs. It follows that every Gallai‐colored K_n contains a monochromatic double star with at least 3n+ 1/4 vertices, a monochromatic complete bipartite graph on at least n/2 vertices, monochromatic subgraphs of diameter two with at least 3n/4 vertices, etc. The generalizations are not automatic though, for instance, a Gallai‐colored complete graph does not necessarily contain a monochromatic star on n/2 vertices. It turns out that the extension is possible for graph classes closed under a simple operation called equalization. We also investigate Ramsey numbers of graphs in Gallai‐colorings with a given number of colors. For any graph H let RG(r, H) be the minimum m such that in every Gallai‐coloring of K_m with r colors, there is a monochromatic copy of H. We show that for fixed H, RG (r, H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if H is a star (and we determine its value). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 233–243, 2010     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Upper bounds on probability thresholds for asymmetric Ramsey properties

Given two graphs G and H, we investigate for which functions the random graph (the binomial random graph on n vertices with edge probability p) satisfies with probability that every red‐blue‐coloring of its edges contains a red copy of G or a blue copy of H. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which G and H are complete graphs of arbitrary size. In our proof we present an alternative to the so‐called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case G = H), and has been used in many proofs of similar results since then.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 1–28, 2014     

A note on the monotonicity of mixed Ramsey numbers

For two graphs, G and H, an edge coloring of a complete graph is (G,H)-good if there is no monochromatic subgraph isomorphic to G and no rainbow subgraph isomorphic to H in this coloring. The set of numbers of colors used by (G,H)-good colorings of K_n is called a mixed Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is “yes” if G is not a star and H does not contain a pendant edge.     

Archimedean ϕ ‐tolerance graphs

Let ? be a symmetric binary function, positive valued on positive arguments. A graph G = (V,E) is a ?‐tolerance graph if each vertex υ ∈ V can be assigned a closed interval I_υ and a positive tolerance t_υ so that xy ∈ E ? | I_x ∩ I_y|≥ ? (t_x,t_y). An Archimedean function has the property of tending to infinity whenever one of its arguments tends to infinity. Generalizing a known result of [15] for trees, we prove that every graph in a large class (which includes all chordless suns and cacti and the complete bipartite graphs K_2,k) is a ?‐tolerance graph for all Archimedean functions ?. This property does not hold for most graphs. Next, we present the result that every graph G can be represented as a ?_G‐tolerance graph for some Archimedean polynomial ?_G. Finally, we prove that there is a ?universal”? Archimedean function ? ^* such that every graph G is a ?^*‐tolerance graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 179–194, 2002     

The multi‐color Ramsey number of an odd cycle

Let r_k(G) be the k‐color Ramsey number of a graph G. It is shown that for k?2 and that r_k(C_{2m+ 1})?(c^kk!)^1/m if the Ramsey graphs of r_k(C_{2m+ 1}) are not far away from being regular. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 324–328, 2009