376 research outputs found

    More on Rainbow Cliques in Edge-Colored Graphs

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    In an edge-colored graph GG, a rainbow clique KkK_k is a kk-complete subgraph in which all the edges have distinct colors. Let e(G)e(G) and c(G)c(G) be the number of edges and colors in GG, respectively. In this paper, we show that for any ε>0\varepsilon>0, if e(G)+c(G)≥(1+k−3k−2+2ε)(n2)e(G)+c(G) \geq (1+\frac{k-3}{k-2}+2\varepsilon) {n\choose 2} and k≥3k\geq 3, then for sufficiently large nn, the number of rainbow cliques KkK_k in GG is Ω(nk)\Omega(n^k). We also characterize the extremal graphs GG without a rainbow clique KkK_k, for k=4,5k=4,5, when e(G)+c(G)e(G)+c(G) is maximum. Our results not only address existing questions but also complete the findings of Ehard and Mohr (Ehard and Mohr, Rainbow triangles and cliques in edge-colored graphs. {\it European Journal of Combinatorics, 84:103037,2020}).Comment: 16page

    Algorithms and Bounds for Very Strong Rainbow Coloring

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    A well-studied coloring problem is to assign colors to the edges of a graph GG so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in kk and the treewidth of GG. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor n1−εn^{1-\varepsilon}, unless P==NP

    Properly coloured copies and rainbow copies of large graphs with small maximum degree

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    Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz local lemma to show the following two results about colourings c of the edges of the complete graph K_n. If for each vertex v of K_n the colouring c assigns each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a copy of G in K_n which is properly edge-coloured by c. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2 edges of K_n, then there is a copy of G in K_n such that each edge of G receives a different colour from c. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page

    On small Mixed Pattern Ramsey numbers

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    We call the minimum order of any complete graph so that for any coloring of the edges by kk colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph HH with edges colored from the above set of kk colors, if we consider the condition of excluding HH in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted Mk(H)M_k(H). We determine this function in terms of kk for all colored 44-cycles and all colored 44-cliques. We also find bounds for Mk(H)M_k(H) when HH is a monochromatic odd cycles, or a star for sufficiently large kk. We state several open questions.Comment: 16 page
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