Minimum size of n-factor-critical graphs and k-extendable graphs

TL;DR

This paper establishes the minimum size of n-factor-critical and k-extendable graphs, including bipartite and non-bipartite cases, using Harary graphs and related structures, and proposes a conjecture for general k.

Contribution

It determines the minimum sizes for n-factor-critical and k-extendable graphs, extending known results and proposing a new conjecture for broader cases.

Findings

Minimum size of n-factor-critical graphs identified

Minimum size of k-extendable bipartite graphs determined

Conjecture proposed for k-extendable non-bipartite graphs

Abstract

We determine the minimum size of $n$-factor-critical graphs and that of $k$-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of $k$-extendable non-bipartite graphs for $k=1,\ 2$, and pose a related conjecture for general $k$.