Maximal 1-plane graphs with the maximum number of crossings

A drawing of a graph in the plane is called 1-planar if each edge is crossed at most once. A graph together with a 1-planar drawing is a 1-plane graph. A 1-plane graph $G$ with exactly $4|V (G)|-8$ edges is called optimal. The crossing number $cr(G)$ of a graph $G$ is the minimum number of crossings over all drawings of $G$. Czap and Hud\'{a}k proved that $cr(G)\le |V(G)|-2$ for any 1-plane graph $G$ and equality holds if $G$ is an optimal 1-plane graph [The Electronic J. Comb}., 20(2),#P54 (2013)]. This paper aims to characterize maximal 1-plane graphs $G$ achieving the maximum crossing number $|V(G)|-2$. We first introduce a class of quasi-optimal 1-plane graphs as a generalization of optimal 1-plane graphs, and then prove that for any maximal 1-plane graph $G$, $cr(G)=|V(G)|-2$ holds if and only if $G$ is a quasi-optimal 1-plane graph. Moreover, we prove that every quasi-optimal 1-plane graph is maximal 1-planar (not merely drawing-saturated). Finally, we present some applications of our main results, including a disproof of an upper bound on the crossing number of maximal 1-planar graphs with odd-degree vertices.