Diana Sasaki Simone Dantas Celina

Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Conjecture. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Several families of snarks have had their total chromatic number determined to be 4, such as the Flower Snark family, the Goldberg family and the Loupekhine family. We show properties of 4-total colorings of the dot product, of the square product, and of the star product, known operations for constructing snarks. We consider subfamilies of snarks using the star product and obtain a 4-total coloring for each family. Moreover, we prove a property about a specific 4-total coloring of both Blanusa subfamilies, two specific families constructed by the dot product of Petersen graphs.