9,064 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

    Partitioning Perfect Graphs into Stars

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    The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor

    Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

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    A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.Comment: 12 pages, 12 figure
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