On the Thickness of Graphs with Genus $2$

Abstract

By a graph we mean an undirected simple graph. The genus $\gamma (G)$ of a graph $G$ is the minimum genus of the orientable surface on which $G$ is embeddable. The thickness $\mathrm{\Theta }(G)$ of $G$ is the minimum number of planar subgraphs whose union is $G$.

In [1], it is proved that, if $\gamma (G)=1$, then $\mathrm{\Theta }(G)=2$. If $\gamma (G)=2$, the known best upper bound on $\mathrm{\Theta }(G)$ is $4$ and, as far as the author knows, the known best lower bound is $2$. In this paper, we prove that, if $\gamma (G)=2$, then $\mathrm{\Theta }(G)\le 3$.