On the Thickness of Graphs with Genus \(2\)

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 \(\Theta(G)\) of \(G\) is the minimum number of planar subgraphs whose union is \(G\).

In [1], it is proved that, if \(\gamma(G) = 1\), then \(\Theta(G) = 2\). If \(\gamma(G) = 2\), the known best upper bound on \(\Theta(G)\) 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 \(\Theta(G) \leq 3\).