A Note on Acyclic Total Coloring of Plane Graphs

Abstract

An acyclic total coloring is a proper total coloring of a graph \(G\) such that there are at least \(4\) colors on vertices and edges incident with a cycle of \(G\). The acyclic total chromatic number of \(G\), \(\chi”_a(G)\), is the least number of colors in an acyclic total coloring of \(G\). In this paper, we prove that for every plane graph \(G\) with maximum degree \(\Delta\) and girth \(g(G)\), \(\chi_a(G) = \Delta+1\) if (1) \(\Delta \geq 9\) and \(g(G) \geq 4\); (2) \(\Delta \geq 6\) and \(g(G) \geq 5\); (3) \(\Delta \geq 4\) and \(g(G) \geq 6\); (4) \(\Delta \geq 3\) and \(g(G) \geq 14\).