18 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 Geometric Graphs

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    Let X1,X2,,XnX_1,X_2,\ldots,X_n be chosen independently and uniformly at random from the unit dd-dimensional cube [0,1]d[0,1]^d. Let rr be given and let X={X1,X2,,Xn}\cal X=\{X_1,X_2,\ldots,X_n\}. The random geometric graph G=GX,rG=G_{\cal X,r} has vertex set X\cal X and an edge XiXjX_iX_j whenever XiXjr\|X_i-X_j\|\leq r. We show that if each edge of GG is colored independently from one of n+o(n)n+o(n) colors and rr has the smallest value such that GG has minimum degree at least two, then GG contains a rainbow Hamilton cycle a.a.s

    On rainbow thresholds

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    Resolving a recent problem of Bell, Frieze, and Marbach, we establish the threshold result of Frankston--Kahn--Narayanan--Park in the rainbow setting.Comment: 10 page

    Random cliques in random graphs

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    We show that for each r4r\ge 4, in a density range extending up to, and slightly beyond, the threshold for a KrK_r-factor, the copies of KrK_r in the random graph G(n,p)G(n,p) are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus, an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem -- recently announced by Jeff Kahn -- implies a corresponding bound for the threshold for G(n,p)G(n,p) to contain a KrK_r-factor. We also prove a slightly weaker result for r=3r=3, and (weaker) generalizations replacing KrK_r by certain other graphs FF. As an application of the latter we find, up to a log factor, the threshold for G(n,p)G(n,p) to contain an FF-factor when FF is 11-balanced but not strictly 11-balanced.Comment: 19 pages; expanded introduction and Section 5, plus minor correction

    Rainbow Thresholds

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    We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include rainbow versions
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