We give a constructive characterization of undirected graphs which contain k spanning trees after adding any new edge. This is a generalization of a theorem of Henneberg and Laman who gave the characterization for k = 2. We also give a constructive characterization of graphs which have k edgedisjoint spanning trees after deleting any edge of them.
As a common generalization of matchings and matroid intersection, Cunningham and Geelen introduced the notion of path-matchings, then they introduced the more general notion of even factor in weakly symmetric digraphs. Here we give a min-max formula for the maximum cardinality of an even factor. Our proof is purely combinatorial. We also provide a Gallai-Edmonds-type structure theorem for even factors.
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