On the restricted matching of graphs in surfaces

TL;DR

This paper investigates the matching extension properties of graphs embedded on surfaces, generalizing previous results and establishing new bounds on when such graphs lack certain extension properties.

Contribution

It extends prior work by proving that no graphs embedded on any surface have property E(m,n) under certain conditions, and shows that large enough graphs are not k-extendable.

Findings

No graphs embedded on any surface have property E(m,1) for certain parameters.

Large embedded graphs lack property E(k-1,1) for all k≥4.

Results include a new proof for the case k=4, aligning with recent findings.

Abstract

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap F=\varnothing$. In particular, a graph with $E(m,0)$ is $m$-extendable. Let $\mu(\Sigma)$ be the smallest integer $k$ such that no graphs embedded on a surface $\Sigma$ are $k$-extendable. Aldred and Plummer have proved that no graphs embedded on the surfaces $\Sigma$ such as the sphere, the projective plane, the torus, and the Klein bottle are $E(\mu(\Sigma)-1,1)$. In this paper, we show that this result always holds for any surface. Furthermore, we obtain that if a graph $G$ embedded on a surface has sufficiently many vertices, then $G$ has no property $E(k-1,1)$ for each integer $k\geq 4$, which implies that $G$ is not $k$-extendable. In the case of $k=4$, we get immediately a main result that Aldred et al. recently obtained.