A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Some Cubic Graphs

Abstract

An adjacent vertex distinguishing total coloring of a graph $G$ is a proper total coloring of $G$ such that no two adjacent vertices are incident to the same set of colors. The minimum number of colors needed for such a coloring is denoted by ${\chi }_{at}(G)$. In this note, we prove that ${\chi }_{at}(G)=5$ for some cubic graphs.