178 research outputs found

    Rainbow Matchings and Hamilton Cycles in Random Graphs

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    Let HPn,m,kHP_{n,m,k} be drawn uniformly from all kk-uniform, kk-partite hypergraphs where each part of the partition is a disjoint copy of [n][n]. We let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge randomly from one of \k colors. We show that if \k=n and m=Knlognm=Kn\log n where KK is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if nn is even and m=Knlognm=Kn\log n where KK is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in Gn,m(n)G^{(n)}_{n,m}. Here Gn,m(n)G^{(n)}_{n,m} denotes a random edge coloring of Gn,mG_{n,m} with nn colors. When nn is odd, our proof requires m=\om(n\log n) for there to be a rainbow Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph

    Rainbow Hamilton cycles in random regular graphs

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    A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page

    Bounded colorings of multipartite graphs and hypergraphs

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    Let cc be an edge-coloring of the complete nn-vertex graph KnK_n. The problem of finding properly colored and rainbow Hamilton cycles in cc was initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruci\'nski. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph GG. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring cc of the complete balanced mm-partite graph to contain a properly colored or rainbow copy of a given graph GG with maximum degree Δ\Delta. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes Δm\Delta \gg m and Δm\Delta \ll m Our main tool is the framework of Lu and Sz\'ekely for the space of random bijections, which we extend to product spaces

    Minimum degree conditions for monochromatic cycle partitioning

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    A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any rr-edge-coloured complete graph has a partition into O(r2logr)O(r^2 \log r) monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant cc such that any rr-edge-coloured graph on nn vertices with minimum degree at least n/2+crlognn/2 + c \cdot r \log n has a partition into O(r2)O(r^2) monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.Comment: 22 pages (26 including appendix
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