Extending Matchings in Planar Odd Graphs

Abstract

A graph $G$ with $1\le n\le |V(G)|–2$ is said to be $n$-factor-critical if any $n$ vertices of $G$ are deleted, then the resultant graph has a perfect matching. An odd graph $G$ with $2k\le |V(G)|–3$ is said to be near $k$-extendable if $G$ has a $k$-matching and any $k$-matching of $G$ can be extended to a near perfect matching of $G$. Lou and Yu [Australas. J. Combin. 29 (2004) 127-133] showed that any $5$-connected planar odd graph is $3$-factor-critical. In this paper, as an improvement of Lou and Yu’s result, we prove that any $4$-connected planar odd graph is $3$-factor-critical and also near $2$-extendable. Furthermore, we prove that all $5$-connected planar odd graphs are near $3$-extendable.