The Adjacent Vertex Distinguishing Total Chromatic Number of Some Families of Snarks

Abstract

The adjacent vertex distinguishing total chromatic number ${\chi }_{at}(G)$ of a graph $G$ is the smallest integer $k$ for which $G$ admits a proper $k$-total coloring such that no pair of adjacent vertices are incident to the same set of colors. Snarks are connected bridgeless cubic graphs with chromatic index $4$. In this paper, we show that ${\chi }_{at}(G)=5$ for two infinite subfamilies of snarks, i.e., the Loupekhine snark and Blanusa snark of first and second kind. In addition, we give an adjacent vertex distinguishing total coloring using $5$ colors for Watkins snark and Szekeres snark, respectively.