On restricted matching extension of 1-embeddable graphs in surfaces with small genus

Let G be a connected graph with at least $2(m+n+1)$ vertices that contains a perfect matching. Then G is $E(m,n)$ if for each pair of disjoint matchings $M,N\subseteq E\left(G\right)$ of size m and n, respectively, there exists a perfect matching F in G such that $M\subseteq F$ and $F\cap N=\varnothing$. A graph G is 1-embeddable in a surface Σ if G can be drawn in Σ so that every edge of G crosses at most one other edge at a point. R.E.L. Aldred and M.D. Plummer [1], [2] investigated the properties $E(m,n)$ for graphs embedded in the plane, torus, projective plane and Klein bottle. In this paper, we study the property $E(m,n)$ for 1-embeddable graphs in surfaces with small genus. It is shown that no 1-embeddable graph in the plane or projective plane is $E(4,1)$ and no 1-embeddable graph in the torus or Klein bottle is $E(5,1)$. As corollaries, no 1-embeddable graph in the plane or projective plane is 5-extendable and no 1-embeddable graph in the torus or Klein bottle is 6-extendable. Some examples show that such results are best possible.