Structural Remarks on Bipartite Graphs with Unique f-Factors
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In this note we will derive some structural results for a bipartite graph G with a unique f-factor. Two necessary conditions will be that G is saturated, meaning that the addition of any edge leads to a second f-factor, and that fA, fB≥1. Here fA and fB are defined as the minimum of f over the vertices in the two partite sets A and B of G, respectively. Our main result states that G has at least fA + fB vertices for which dG (v) = f(v) holds.
KeywordsUnique f-factor Bipartite graphs Extremal graphs
Mathematics Subject ClassificiationPrimary 05C70 Secondary 05C35
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