5,854 research outputs found
The Cost of Perfection for Matchings in Graphs
Perfect matchings and maximum weight matchings are two fundamental
combinatorial structures. We consider the ratio between the maximum weight of a
perfect matching and the maximum weight of a general matching. Motivated by the
computer graphics application in triangle meshes, where we seek to convert a
triangulation into a quadrangulation by merging pairs of adjacent triangles, we
focus mainly on bridgeless cubic graphs. First, we characterize graphs that
attain the extreme ratios. Second, we present a lower bound for all bridgeless
cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic
graphs, most of which are shown to be tight. Additionally, we present tight
bounds for the class of regular bipartite graphs
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
Efficient edge domination in regular graphs
An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set
of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced
matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and
that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A
necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove
that, for arbitrary fixed p 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complet
Determination of (0,2)-regular sets in graphs and applications
In this paper, relevant results about the determination of (k,t)-regular sets, using the main eigenvalues of a graph, are reviewed and some results about the determination of (0,2)-regular sets are introduced. An algorithm for that purpose is also described. As an illustration, this algorithm is applied to the determination of maximum matchings in arbitrary graphs
Bicolored Matchings in Some Classes of Graphs
We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R| = n + 1 such that perfect matchings with k red edges exist for all k, 0 ≤ k ≤ n. Given two integers p < q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p ≤ 4 there is a set R with |R| = p for which perfect matchingsMk exist with |Mk ∩R| ≤ k for all k ≤ p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k
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