Binding Number, Toughness and General Matching Extendability in Graphs

TL;DR

This paper establishes new conditions based on binding number and toughness for large graphs to guarantee their extendability properties related to perfect matchings, generalizing classical matching theory results.

Contribution

It introduces novel bounds involving binding number and toughness that ensure graphs are extendable for perfect matchings, extending prior matching existence criteria.

Findings

Graphs with large enough order and binding number ≥ 4/3+ε are extendable.

Graphs with high toughness and connectivity are also extendable.

Conditions for extendability closely match those for the existence of perfect matchings.

Abstract

A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect matching $F$ in $G$ such that $M\subseteq F$ and $N\cap F=\emptyset$. Similarly, a connected graph with at least $n+2k+2$ vertices is called $(n,k)$-{\it extendable} if for any vertex set $S$ of size $n$ and any matching $M$ of size $k$ of $G-S$, $G-S-V(M)$ contains a perfect matching. Let $\varepsilon$ be a small positive constant, $b(G)$ and $t(G)$ be the binding number and toughness of a graph $G$. The two main theorems of this paper are: for every graph $G$ with sufficiently large order, 1) if $b(G)\geq 4/3+\varepsilon$, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable; 2) if $t(G)\geq 1+\varepsilon$ and $G$ has a high connectivity, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings.