On the restricted matching of graphs in surfaces

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 $μ(Σ)$ be the smallest integer $k$ such that no graphs embedded on a surface $Σ$ are $k$-extendable. Aldred and Plummer have proved that no graphs embedded on the surfaces $Σ$ such as the sphere, the projective plane, the torus, and the Klein bottle are $E(μ(Σ)-1,1)$. 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.

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