On total coloring and equitable total coloring of infinite snark families

Abstract

We show that all members of the SemiBlowup, Blowup and the first Loupekine snark families have equitable total chromatic number equal to 4. These results provide evidence of negative answers for the questions proposed: by Cavicchioli et al. (2003) about the smallest order of a Type 2 snark of girth at least 5; and by Dantas et al. (2016) about the existence of Type 1 cubic graph with girth at least 5 and equitable total chromatic number 5. Moreover, we show new infinite families of snarks obtained by the Kochol superpositions that are Type 1.