On the Number of Group-Weighted Matchings

Authors:
Jeff Kahn;Roy Meshulam
Affiliations:
Dept. of Mathematics and RUTCOR, Rutgers University, New Brunswick, NJ 08903;Dept. of Mathematics, Technion, Haifa 32000, Israel
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
1998

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Abstract

Let G be a bipartite graph with a bicoloration {A,B},|A|=|B|. Let E(G) ⊆ A x B denote the edge set of G, and let m(G) denotethe number of perfect matchings of G. Let K be a (multiplicative) finiteabelsian group |K| = k, and let w:E(G) → K be a weight assignment onthe edges of G. FOr S ⊆ E(G) let w(S) =∏_e∈Sw(e). A perfect matching M of G is a w-matchingif w(M)=1. We shall be interested in m(G,w), the number of w-matchings of G.It is shown that if deg(a) ≥ d for all a ∈ A, then either G hasno w-matchings, or G has at least (d - k + 1)! w-matchings.