Characterizing minimally $n$-extendable bipartite graphs

Abstract

In this paper, it is proved that let $G$ be a bipartite graph with bipartition $(X,Y)$ and with a perfect matching $M$, let $G$ be an $n$-extendable graph, then $G$ is minimally $n$-extendable if and only if, for any two vertices $x\in X$ and $y\in Y$ such that $\mathit{xy}\in E(G)$, there are exactly $n$ internally disjoint $(x,y)M$-alternating paths $P_1,P_2,\dots ,P_n$ such that $P_i(1⩽i⩽n)$ starts and ends with edges in $E(G)⧹M$.