Extending Matchings in Planar Odd Graphs

Abstract

A graph \(G\) with \(1 \leq n \leq |V(G)| – 2\) 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 \(2k \leq |V(G)| – 3\) 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.