Abstract
A q-total coloring of G is an assignment of q colors to the vertices or edges of G, so that adjacent or incident elements have different colors. The Total Coloring Conjecture (TCC) asserts that a total coloring of a graph G has at least ∆ + 1 and at most ∆ + 2 colors. Rosenfeld has shown that the total chromatic number of a cubic graph is either 4 (Type 1) or 5 (Type 2). We present Type 1 new infinite families of snarks (cubic bridgeless graphs of chromatic index 4) obtained by the Kochol superposition of Goldberg with t-Semiblowup snarks. These results provide evidence of a negative answer for the question proposed by Cavicchioli et al. (2003) about the smallest order of a Type 2 snark of girth at least 5.
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