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 \(\Delta(G)\) and girth \(g\) is at most \(\Delta(G)+2\) if \(\Delta \geq 12\), or \(\Delta \geq 6\) and \(g \geq 4\), or \(\Delta = 5\) and \(g \geq 5\), or \(g \geq 6\). Moreover, if \(G\) is a series-parallel graph with \(\Delta \geq 3\) or a planar graph with \(\Delta \geq 3\) and \(g \geq 12\), then the acyclic total chromatic number of \(G\) is \(\Delta(G) + 1\).