The matching extendability of optimal 2-planar graphs

Abstract

A graph is 2-planar if it can be drawn in the plane such that each edge is crossed by at most two other edges. It is known that for a 2-planar graph $G$, $|E(G)| \le 5|V(G)| - 10$. When the equality holds, we call $G$ an optimal 2-planar graph. This paper investigates the matching extendability of optimal 2-planar graphs. By local optimality, we prove that every 4-connected optimal 2-planar graph $G$ of even order is 1-extendable, and give a criterion for $G$ to be 2-extendable. We also prove that no optimal 2-planar graph is 5-extendable and construct a 4-extendable optimal 2-planar graph based on the dodecahedron. Finally, we show that every 6-connected optimal 2-planar graph of even order with at least $2m+2$ vertices is distance 3 $m$-extendable for any $m \ge 0$.