Type 1 Kochol Superpositions: Goldberg and Loupekine with Blowup and Semiblowup snarks

Giovanna A. B. Penao1, Miguel A. D. R. Palma1,2, Simone Dantas1, Diana Sasaki3
1IME, Universidade Federal Fluminense, Niterói, RJ, 24210-201, Brazil
2CCET, Universidade Federal do Maranhão, São Luís, MA, 65080-805, Brazil
3IME, Universidade do Estado do Rio de Janeiro, Rio de janeiro, RJ, 20550-900, Brazil

Abstract

A \(q\)-total coloring of \(G\) is an assignment of \(q\) colors to the vertices and 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 \(\Delta+1\) and at most \(\Delta+2\) colors. In this paper, we determine that all members of new infinite families of snarks obtained by the Kochol superposition of Goldberg and Loupekine with Blowup and Semiblowup snarks are Type~1. These results contribute to a question posed by Brinkmann, Preissmann and D. Sasaki (2015) by presenting negative evidence about the existence of Type~2 cubic graphs with girth at least 5.

Keywords: Total coloring, Kochol superposition, Snarks