592 research outputs found

    Existences of rainbow matchings and rainbow matching covers

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    Let GG be an edge-coloured graph. A rainbow subgraph in GG is a subgraph such that its edges have distinct colours. The minimum colour degree δc(G)\delta^c(G) of GG is the smallest number of distinct colours on the edges incident with a vertex of GG. We show that every edge-coloured graph GG on n7k/2+2n\geq 7k/2+2 vertices with δc(G)k\delta^c(G) \geq k contains a rainbow matching of size at least kk, which improves the previous result for k10k \ge 10. Let Δmon(G)\Delta_{\text{mon}}(G) be the maximum number of edges of the same colour incident with a vertex of GG. We also prove that if t11t \ge 11 and Δmon(G)t\Delta_{\text{mon}}(G) \le t, then GG can be edge-decomposed into at most tn/2\lfloor tn/2 \rfloor rainbow matchings. This result is sharp and improves a result of LeSaulnier and West

    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 matchings in bipartite multigraphs

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    Suppose that kk is a non-negative integer and a bipartite multigraph GG is the union of N=k+2k+1n(k+1)N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1) matchings M1,,MNM_1,\dots,M_N, each of size nn. We show that GG has a rainbow matching of size nkn-k, i.e. a matching of size nkn-k with all edges coming from different MiM_i's. Several choices of parameters relate to known results and conjectures
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