Hall Conditions for Edge-weighted Bipartite Graphs

A weighted variant of Hall’s condition for the existence of matchings is shown to be equivalent to the existence of a matching in a lexicographic product. This is used to introduce characterizations of those bipartite graphs whose edges may be replicated so as to yield semiregular multigraphs or, equivalently, semiregular edge-weightings. Such bipartite graphs will be called semiregularizable. For example, Hall’s theorem guarantees the existence of a spanning (1, 1)-semiregularizable forest (i.e., a perfect matching) while a p/q rational variant of Hall’s theorem guarantees a spanning (p, q)-semiregularizable forest. Some infinite families of semiregularizable trees are described and all semiregularizable trees on at most 11 vertices are listed. Matrix analogues of some of the results are mentioned and are shown to imply some of the known characterizations of regularizable graphs.