361 research outputs found
A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs
We show that the number of -matching in a given undirected graph
is equal to the number of perfect matching of the corresponding graph
on an even number of vertices divided by a suitable factor.
If is bipartite then one can construct a bipartite .
For bipartite graphs this result implies that the number of -matching has
a polynomial-time approximation algorithm. The above results are extended to
permanents and hafnians of corresponding matrices.Comment: 6 page
Rainbow perfect matchings in r-partite graph structures
A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft
Strong games played on random graphs
In a strong game played on the edge set of a graph G there are two players,
Red and Blue, alternating turns in claiming previously unclaimed edges of G
(with Red playing first). The winner is the first one to claim all the edges of
some target structure (such as a clique, a perfect matching, a Hamilton cycle,
etc.). It is well known that Red can always ensure at least a draw in any
strong game, but finding explicit winning strategies is a difficult and a quite
rare task. We consider strong games played on the edge set of a random graph G
~ G(n,p) on n vertices. We prove, for sufficiently large and a fixed
constant 0 < p < 1, that Red can w.h.p win the perfect matching game on a
random graph G ~ G(n,p)
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , we determine the
number of random edges required to add to an arbitrary graph of linear minimum
degree in order to ensure the resulting graph contains a perfect -tiling
with high probability. Our proof utilises Szemer\'edi's Regularity lemma as
well as a special case of a result of Koml\'os concerning almost perfect
-tilings in dense graphs.Comment: 19 pages, to appear in CP
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