The thickness of a graph with constraint on girth

Abstract

The thickness of a graph $G$, denoted by $\theta \left(G\right)$, is the minimum integer $t$ such that $G$ can be decomposed into $t$ planar subgraphs. In 1991, Dean et al. conjectured that the thickness of a graph $G$ with $m$ edges is at most $\sqrt{m/16}+O\left(1\right)$. In this paper we show that the conjecture holds if the girth $g$ of $G$ is at least five, and that $\theta \left(G\right)<1+\sqrt[s]{m/2}$ if $g\ge 6$, where $s=\lfloor g/2\rfloor$. If the girth of $G$ is not restricted, then we prove that $\theta \left(G\right)\le \lfloor 5/6+\sqrt{2m/9-23/36}\rfloor$ where $m\ge 3$, which improves the known bound for the thickness of graphs. If $G$ is a connected nonplanar graph with girth $g\ge 6$ which is 2-cell embedded in the orientable surface $S_k$ (the nonorientable surface $N_h$, respectively), then $\theta \left(G\right)\le 3+\lfloor \sqrt[s]{2k-1}\rfloor$ ($\theta \left(G\right)\le 3+\lfloor \sqrt[s]{h-1}\rfloor$,respectively). Clearly, if $k(\ge 1)$ or $h(\ge 2)$ is fixed, then the aforementioned bounds get closer and closer to 4, as the girth $g$ is larger and larger