منابع مشابه
On maximum matchings in almost regular graphs
In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph G with 2 ≤ δ(G) ≤ ∆(G) ≤ 3 contains a maximum matching whose unsaturated vertices do not have a common neighbor, where ∆(G) and δ(G) denote the maximum and minimum degrees of vertices in G, respectively. In the same paper they suggested the following conjecture: every graph G with ∆(G)− δ(G) ≤ 1 contains a maximum matching whos...
متن کاملMaximum Matchings in Regular Graphs of High Girth
Let G = (V,E) be any d-regular graph with girth g on n vertices, for d ≥ 3. This note shows that G has a maximum matching which includes all but an exponentially small fraction of the vertices, O((d − 1)−g/2). Specifically, in a maximum matching of G, the number of unmatched vertices is at most n/n0(d, g), where n0(d, g) is the number of vertices in a ball of radius b(g − 1)/2c around a vertex,...
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For a simple d-regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for x ∈ V (G) let p(x) be the probability that M does not cover x. We show that for large d, the p(x)’s and the mean μ and variance σ of |M | are determined to within small tolerances just by d and (in the case of μ and σ) |V (G)|: Theorem For any d-regular graph G, (a) p(x) ∼ d ∀x ∈ V (G...
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We provide a formula for the number of edges of a maximum induced matching in a graph. As applications, we give some structural properties of (k + 1 )K2-free graphs, construct all 2K2-free graphs, and count the number of labeled 2K2-free connected bipartite graphs.
متن کاملPerfect matchings in regular graphs from eigenvalues
Throughout, G denotes a simple graph of order n (the number of vertices) and size e (the number of edges). The eigenvalues of G are the eigenvalues λi 1 Research partially supported by an NSERC postdoctoral fellowship. 2 Research supported by the Natural Sciences and Engineering Research Council of Canada. Email addresses: [email protected] (Sebastian M. Cioabă), [email protected] (D...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2018
ISSN: 0012-365X
DOI: 10.1016/j.disc.2018.01.016