Acyclic Total Colorings of Planar Graphs

Abstract

In this paper, we give the definition of acyclic total coloring and acyclic total chromatic number of a graph. It is proved that the acyclic total chromatic number of a planar graph $G$ with maximum degree $\mathrm{\Delta }(G)$ and girth $g$ is at most $\mathrm{\Delta }(G)+2$ if $\mathrm{\Delta }\ge 12$, or $\mathrm{\Delta }\ge 6$ and $g\ge 4$, or $\mathrm{\Delta }=5$ and $g\ge 5$, or $g\ge 6$. Moreover, if $G$ is a series-parallel graph with $\mathrm{\Delta }\ge 3$ or a planar graph with $\mathrm{\Delta }\ge 3$ and $g\ge 12$, then the acyclic total chromatic number of $G$ is $\mathrm{\Delta }(G)+1$.