605 research outputs found

    K3K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

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    A K3K_3-WORM coloring of a graph GG is an assignment of colors to the vertices in such a way that the vertices of each K3K_3-subgraph of GG get precisely two colors. We study graphs GG which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a K3K_3-WORM coloring with two colors. In fact for every integer k3k\ge 3 there exists a K3K_3-WORM colorable graph in which the minimum number of colors is exactly kk. There also exist K3K_3-WORM colorable graphs which have a K3K_3-WORM coloring with two colors and also with kk colors but no coloring with any of 3,,k13,\dots,k-1 colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide kk-colorability for every k2k \ge 2 (and remains intractable even for graphs of maximum degree 9 if k=3k=3). On the other hand, we prove positive results for dd-degenerate graphs with small dd, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.Comment: 18 page

    On two problems in Ramsey-Tur\'an theory

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    Alon, Balogh, Keevash and Sudakov proved that the (k1)(k-1)-partite Tur\'an graph maximizes the number of distinct rr-edge-colorings with no monochromatic KkK_k for all fixed kk and r=2,3r=2,3, among all nn-vertex graphs. In this paper, we determine this function asymptotically for r=2r=2 among nn-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an nn-vertex KkK_k-free graph GG with α(G)=o(n)\alpha(G)=o(n). The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 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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