The Matching Extendability of 7-Connected Maximal 1-Plane Graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. A graph is said to be k(>= 1)-extendable if every matching of size k can be extended to a perfect matching. It is known that the vertex connectivity of a 1-plane graph is at most 7. In this paper, we characterize the k-extendability of 7-connected maximal 1-plane graphs. We show that every 7-connected maximal 1-plane graph with even order is k-extendable for 1 <= k <= 3. And any 7-connected maximal 1-plane graph is not k-extendable for 4 <= k <= 11. As for k >= 12, any 7-connected maximal 1-plane graph with n vertices is not k-extendable unless n = 2k.