Abstract
Let G be a bipartite graph with bicoloration {A, B}, |A| = |B|, 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) = 1.
A characterization is given for all w's for which every perfect matching is a w-matching.
It is shown that if G = K k+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 \(\in\) A, then either G has no w-matchings, or G has at least (d − 1)! w-matchings.
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R. Meshulam: Research supported by a Technion V.P.R. Grant No. 100-854.
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Aharoni, R., Meshulam, R. & Wajnryb, B. Group Weighted Matchings in Bipartite Graphs. Journal of Algebraic Combinatorics 4, 165–171 (1995). https://doi.org/10.1023/A:1022433630971
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DOI: https://doi.org/10.1023/A:1022433630971