A Novel Characterization of $n$-Extendable Bipartite Graphs

Abstract

Let $G$ be a simple connected graph. For a subset $S$ of $V(G)$ with $|S|=2n+1$, let ${\alpha }_{(2n+1)}(G,S)$ denote the graph obtained from $G$ by contracting $S$ to a single vertex. The graph ${\alpha }_{(2n+1)}(G,S)$ is also said to be obtained from $G$ by an ${\alpha }_{(2n+1)}$-contraction. For pairwise disjoint subsets $S_1,S_2,\dots ,S_{2n}$ of $V(G)$, let ${\beta }_n(G,S_1,S_2,\dots ,S_{2n})$ denote the graph obtained from $G$ by contracting each $S_i$ ($i=1,2,\dots ,2n$) to a single vertex respectively. The graph ${\beta }_{2n}(G,S_1,S_2,\dots ,S_{2n})$ is also said to be obtained from $G$ by a ${\beta }_{2n}$-contraction. In the present paper, based on ${\alpha }_{(2n+1)}$-contraction and ${\beta }_2$-contraction, some new characterizations for $n$-extendable bipartite graphs are given.