Equivalence between Extendibility and Factor-Criticality

TL;DR

This paper establishes a precise equivalence between extendibility and factor-criticality in graphs, providing bounds that are proven to be optimal, thereby addressing a problem posed by Favaron and Yu.

Contribution

It proves new necessary and sufficient conditions linking k-extendability and factor-criticality in graphs, with optimal bounds.

Findings

Established equivalence between k-extendability and 2k-factor-criticality for certain bounds.

Proved the bounds are tight with counterexamples.

Addressed and solved a problem posed by Favaron and Yu.

Abstract

In this paper, we show that if $k\geq (\nu+2)/4$, 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 $k\geq (\nu-3)/4$, a graph $G$ is $k\ 1/2$-extendable if and only if it is $(2k+1)$-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].