Characterization of strong Hall critical graphs

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2010-05-13T21:52:32Z

Authors

Hare, Donovan R.

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Abstract

A bipartite graph G=(X,Y) is strong Hall with respect to Y if for all S \subset Y, S \ne \emptyset, |S|<|N_G(S)|. A bipartite graph G=(X,Y) is strong Hall critical with respect to Y if it is strong Hall with respect to Y but it is no longer once any edge is removed. In this paper strong Hall critical graphs are shown to be characterized by those strong Hall bipartite graphs G=(X,Y) whose vertices in Y have only degree two. It is also shown that strong Hall graphs G=(X,Y) have the property that between any two vertices in X which are connected in G, there is a path between them and a matching which saturates the remaining vertices in Y.

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