Abstract
Let G be a bipartite graph with bicoloration lA, Br, vAv e vBv, and let w : E(G) → K where K is a finite abelian group with k elements. For a subset S ⊂ E(G) let w(S)=\prod _{e \in S} w(e). A Perfect matching M ⊂ E(G) is a w-matching if w(M) e 1. A characterization is given for all w's for which every perfect matching is a w-matching. It is shown that if G e Kk+1,k+1 then either G has no w-matchings or it has at least 2 w-matchings. If K is the group of order 2 and deg(a) ≥ d for all a ∈ A, then either G has no w-matchings, or G has at least (d − 1)e w-matchings.
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