Bipartite graphs with a perfect matching and digraphs

Abstract

In this paper, we introduce a corresponding between bipartite graphs with a perfect matching and digraphs, which implicates an equivalent relation between the extendibility of bipartite graphs and the strongly connectivity of digraphs. Such an equivalent relation explains the similar results on $k$-extendable bipartite graphs and $k$-strong digraphs. We also study the relation among $k$-extendable bipartite graphs, $k$-strong digraphs and combinatorial matrices. For bipartite graphs that are not 1-extendable and digraphs that are not strong, we prove that the elementary components and strong components are counterparts.

2 figures, 7 pages. A short version of this paper is published in "Advances and applications in Discrete Mathematics". This version contains some interesting comparison between $k$-strong digraphs and $k$-extendable bigraphs, which is deleted in the published version