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Rainbow Generalizations of Ramsey Theory: A Survey
Abstract
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.
KeywordsEdge-coloring-Ramsey theory-Rainbow-Heterochromatic-Multicolored-Anti-Ramsey
... To learn more about the Gallai-Ramsey number, please see [18,19]. ...
This paper explores various Ramsey numbers associated with cycles with pendant edges, including the classical Ramsey number, the star-critical Ramsey number, the Gallai–Ramsey number, and the star-critical Gallai–Ramsey number. These Ramsey numbers play a crucial role in combinatorial mathematics, determining the minimum number of vertices required to guarantee specific monochromatic substructures. We establish upper and lower bounds for each of these numbers, providing new insights into their behavior for cycles with pendant edges—graphs formed by attaching additional edges to one or more vertices of a cycle. The results presented contribute to the broader understanding of Ramsey theory and serve as a foundation for future research on generalized Ramsey numbers in complex graph structures.
... This term was introduced by Gyárfás and Simonyi [16] due to the close connection of these colourings to the seminal work of Gallai [12] on comparability graphs, where he obtained a structural classification of all colourings of the complete graph that avoid rainbow triangles. Since its introduction, Gallai colourings have garnered significant attention and have been explored in various contexts, including Ramsey-type problems [6,7,10,14,15], extremal graph theory [3,8,9], and graph entropy [19]. ...
For , we say that a colouring of is t- if no two rainbow t-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of clique counts in random graphs, we study the maximum number of rainbow t-cliques in an almost t-Gallai colouring of . For every , we show that . For t=3, surprisingly, the behaviour is substantially different. Our main result establishes that which gives the first non-trivial improvements over the simple lower and upper bounds. Our proof combines various applications of the probabilistic method and a generalisation of the edge-isoperimetric inequality for the hypercube.
... Since then this branch of combinatorics, especially the mentioned special case, has been flourishing. Several excellent surveys were dedicated to the subject, see [8][9][10]14]. ...
We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order n is equivalent to a proper edge-coloring of . A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined l(n) as the least integer such that every properly edge-colored , which contains at least l(n) different colors, admits a multicolored perfect matching. They conjectured that if n is large enough. In this note we prove that l(n) is bounded from above by if . We point out a connection to anti-Ramsey problems. We propose a conjecture related to a well-known result by Woolbright and Fu, that every proper edge-coloring of admits a multicolored 1-factor.
... However, at the beginning of the 70s, Erdős, Simonovits and T. Sós [3] started the study of anti-Ramsey theory, that is, the study of regularities that concern all-distinct objects (e.g., a subgraph of an edge-colored graph in which all edges have different colors, often called a rainbow -see [4] for a survey). ...
In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that in every aperiodic uniformly recurrent word, anti-powers of every order begin at every position. We further show that every infinite word avoiding anti-powers of order 3 is ultimately periodic, while there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6.
... Several Ramsey-type results in Gallai-colored graphs have also emerged in the literature (see e.g. [3], [8], [12], [13]), but they mostly focus on finding large monochromatic structures in such colorings. Our main result is the observation that certain proof techniques used by Fox-Grinshpun-Pach [7] for solving the multicolor Erdős-Hajnal conjecture for rainbow triangles also give a partial answer for Loh's question [17], that asks the following: what is the value of f (n, r, s), the maximum number such that every r-coloring of the edges of the transitive tournament on n vertices contains a directed path with at least f (n, r, s) vertices whose edges have at most s colors? ...
Fox--Grinshpun--Pach showed that every 3-coloring of the complete graph on n vertices without a rainbow triangle contains a clique of size which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a 2-colored subgraph with chromatic number at least , and this is best possible. We further show that for fixed positive integers s,r with , every r-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a subgraph that uses at most s colors and has chromatic number at least , and this is best possible. Fox--Grinshpun--Pach previously showed a clique version of this result. As a direct corollary of our result we obtain a generalisation of the celebrated theorem of Erd\H{o}s-Szekeres, which states that any sequence of n numbers contains a monotone subsequence of length at least . We prove that if an r-coloring of the edges of an n-vertex tournament does not contain a rainbow triangle then there is an s-colored directed path on vertices, which is best possible. This gives a partial answer to a question of Loh.
Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number grk(G: H) is defined as the minimum integer N such that for all n ≥ N, every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. Denote grk′(G: H) as the minimum integer N such that for all n ≥ N, every edge-coloring of Kn using at most k colors contains either a rainbow copy of G or a monochromatic copy of H. In this paper, we get some exact values or bounds for grk(P5: H) and grk′(P5: H), where H is a cycle or a book graph. In addition, our results support a conjecture of Li, Besse, Magnant, Wang and Watts in 2020.
The anti‐Ramsey number of an ‐graph is the minimum number of colors needed to color the complete ‐vertex ‐graph to ensure the existence of a rainbow copy of . We establish a removal‐type result for the anti‐Ramsey problem of when is the expansion of a hypergraph with a smaller uniformity. We present two applications of this result. First, we refine the general bound proved by Erdős–Simonovits–Sós, where denotes the family of ‐graphs obtained from by removing one edge. Second, we determine the exact value of for large in cases where is the expansion of a specific class of graphs. This extends results of Erdős–Simonovits–Sós on complete graphs to the realm of hypergraphs.
Ramsey’s theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph KN contains either a red Kn or blue Km as a subgraph. The smallest such N is called the Ramsey number, denoted as R(n,m). In this paper, we reformulate this theorem and present a proof of Ramsey’s theorem that is novel as far as we are aware.
Given two graphs G and H, the {\it rainbow number} rb(G,H) for H with respect to G is defined as the minimum number k such that any k-edge-coloring of G contains a rainbow H, i.e., a copy of H, all of whose edges have different colors. Denote by a matching of size k and the class of all plane triangulations of order n, respectively. In [S. Jendrol, I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158--164], the authors determined the exact values of for and proved that for . In this paper, we improve the upper bounds and prove that for and . Especially, we show that for .
Thesis (Ph. D.)--Western Michigan University, 2000. Includes bibliographical references (leaves 94-95).
Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex colorings. With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.
Given a positive integer n and a family F of graphs, let R*(n, F) denote the maximum number of colors in an edge-coloring of Kn such that no subgraph of Kn belonging to F has distinct colors on its edges. We determine R*(n, Tk) where Tk is the family of trees with k edges. We derive general bounds for R*(n, T) where T is an arbitrary tree with k edges. Finally, we present a single tree T with k edges such that R*(n, T) is nearly as small as R*(n, Tk).
Let f(G) be the maximum number of colors in a vertex coloring of a simple plane graph G such that no face has distinct colors on all its vertices. If G has n vertices and chromatic number k, f(G) ≥ ⌈ n/k ⌉ + 1. For k ∈ {2, 3}, this bound is sharp for all n (except n ≤ 3 when k = 2). For k = 4, the bound is within 1 for all n.
Consider a coloring off1; 2;::: ;ng in 3 colors, where n· 0 (mod 3). If all the color classes have the same cardinality, then there is a 3-term arithmetic progression whose elements are colored in distinct colors. This rainbow variant of van der Waerden's theorem proves the conjecture of the second author.
Given a graph L, in this article we investigate the anti-Ramsey number chi(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, Erdos, Graham, and T Sos asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that chi(s)(n, alpha n(2), L)/n --> infinity as n --> infinity? 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. (C) 2006 Wiley Periodicals, Inc.
The rainbow Ramsey number RR(G 1 ,G 2 ) or constrained Ramsey number f(G 1 ,G 2 ) of two graphs G 1 and G 2 is defined to be the minimum integer N such that any edge-coloring of the complete graph K N with any number of colors must contain either a subgraph isomorphic to G 1 with every edge the same color or a subgraph isomorphic to G 2 with every edge a different color. This number exists if and only if G 1 is a star or G 2 is acyclic. In this paper, we present the conjecture that the constrained Ramsey number of nK 2 and mK 2 is m(n-1)+2, along with a proof in the case m≤3 2(n-1).
Let sr(G, k) be the least m such that in any colouring of the edges of Km using no colour more than k times there is a (isomorphic copy of) G with all edges of distinct colours. Continuing previous work of one of the authors we determine good bounds on sr (K1,s, k) and use them to find exact values for large classes of parameters.