Geodesic
New results on Type 2 snarks
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 879-893

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Snarks are cyclically 4-edge-connected cubic graphs that admit no proper 3-edge-coloring. A snark is of Type 1 if it has a proper total coloring of its vertices and edges with four colors; it is of Type 2 if any total coloring requires at least five colors. Following an extensive computer search, in 2003, Cavicchioli et al. asked whether there exist Type 2 snarks of girth at least 5. This question is still open, however, in 2015, Brinkmann et al. described the first known family of Type 2 snarks of girth 4. In this work we provide new families of Type 2 snarks of girth 4, all of which can be constructed by a dot product of two Type 1 snarks. We also show that the previously constructed Type 2 snarks of Brinkmann et al. do not have this property.
Keywords: dot product, total coloring, snark 
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Dantas, Simone; Marinho, Rodrigo; Preissmann, Myriam; Sasaki, Diana. New results on Type 2 snarks. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 879-893. http://geodesic.mathdoc.fr/item/DMGT_2023_43_4_a0/