On total coloring and equitable total coloring of infinite snark families
Miguel A.D.R. Palma, Isabel F.A. Gonçalves, Diana Sasaki, Simone Dantas · RAIRO - Operations Research · 2023
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 (A. Cavicchioli, T.E. Murgolo, B. Ruini and F. Spaggiari, Acta Appl. Math. 76 (2003) 56–88.) about the smallest order of a Type 2 snark of girth at least 5; and by (S. Dantas, C.M.H. De Figueiredo, G. Mazzuoccolo, M. Preissmann, V.F. dos Santos and D. Sasaki, Discret. Appl. Math. 209 (2016) 84–91.) 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.