On equitable total coloring of snarks

Abstract

The search for counterexamples to the Four Color Conjecture originated snarks, a very special class of cubic graphs. In this paper, we consider the equitable total coloring of snarks. A total coloring is equitable if the number of elements colored with each color differs by at most one, and the least integer for which a graph has such a coloring is called its equitable total chromatic number. In 2002, Wang conjectured that the equitable total chromatic number of a graph is at most ∆ + 2, and this was proved for cubic graphs. Therefore, the equitable total chromatic number of a cubic graph is either 4 or 5. We provide evidence to a negative answer to the question proposed in 2016 about the existence of a Type 1 cubic graph with girth greater than 4 and equitable total chromatic number 5, by determining equitable 4-total colorings for every member of three infinite families of snarks with girth 5.