1,994 research outputs found

    Vertex covers by monochromatic pieces - A survey of results and problems

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    This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

    Vertex covering with monochromatic pieces of few colours

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    In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every 22-colouring of the edges of KnK_n, there is a vertex cover by 2n2\sqrt{n} monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers r,sr,s, what is the smallest number pcr,s(Kn)\text{pc}_{r,s}(K_n) such that in every colouring of the edges of KnK_n with rr colours, there exists a vertex cover of KnK_n by pcr,s(Kn)\text{pc}_{r,s}(K_n) monochromatic paths using altogether at most ss different colours? For fixed integers r>sr>s and as n→∞n\to\infty, we prove that pcr,s(Kn)=Θ(n1/χ)\text{pc}_{r,s}(K_n) = \Theta(n^{1/\chi}), where χ=max⁡{1,2+2s−r}\chi=\max{\{1,2+2s-r\}} is the chromatic number of the Kneser gr aph KG(r,r−s)\text{KG}(r,r-s). More generally, if one replaces KnK_n by an arbitrary nn-vertex graph with fixed independence number α\alpha, then we have pcr,s(G)=O(n1/χ)\text{pc}_{r,s}(G) = O(n^{1/\chi}), where this time around χ\chi is the chromatic number of the Kneser hypergraph KG(α+1)(r,r−s)\text{KG}^{(\alpha+1)}(r,r-s). This result is tight in the sense that there exist graphs with independence number α\alpha for which pcr,s(G)=Ω(n1/χ)\text{pc}_{r,s}(G) = \Omega(n^{1/\chi}). This is in sharp contrast to the case r=sr=s, where it follows from a result of S\'ark\"ozy (2012) that pcr,r(G)\text{pc}_{r,r}(G) depends only on rr and α\alpha, but not on the number of vertices. We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic dd-regular graphs

    Partitioning 3-colored complete graphs into three monochromatic cycles

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    We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible
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