Equivalence between Extendibility and Factor-Criticality

In this paper, we show that if $k\ge (\nu +2)\mathrm{/}4$, where $\nu$ denotes the order of a graph, a non-bipartite graph $G$ is $k$-extendable if and only if it is $2k$-factor-critical. If $k\ge (\nu -3)\mathrm{/}4$, a graph $G$ is $k
1/2$-extendableifandonlyifitis$(2k+1)$-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron [3] and Yu [11].