Coloração total equilibrada dos snarks de Loupekine
Abstract
In 2016, Dantas et al. proposed the question about the existence of a Type 1 cubic graph with girth at least 5 and equitable total chromatic number 5, which motivated our result. We prove that all graphs of the second family of Loupekine snarks have equitable total chromatic number 4, contributing as a negative evidence to the question.References
Cordeiro, L., Dantas, S., and Sasaki, D. (2017). On equitable total colouring of loupekine snarks and their products. Mat. Cont., 45:77–85.
Dantas, S., de Figueiredo, C. M. H., Mazzuoccolo, G., Preissman, M., dos Santos, V. F., and Sasaki, D. (2016). On the equitable total chromatic number of cubic graphs. Discrete Appl. Math., 209:84–91.
Gardner, M. (1976). Mathematical games: Snarks, boojums and other conjectures related to the four-color-map theorem. Sci. Am., pages 126–130.
Isaacs, R. (1976). Loupekhine’s snarks: A bi-family of non-tait-colorable graphs. Tec. Report.
M.Behzad (1965). Graphs and Their Chromatic Numbers. PhD thesis, Michigan State University, Michigan.
Sasaki, D., Dantas, S., de Figueiredo, C. M. H., Mazzuoccolo, G., and Preissman, M. (2014). The hunting of a snark with total chromatic number 5. Discrete Appl. Math., 164:470–481.
Tait, P. G. (1878-1880). Remarks on the colouring of maps. In Proceedings of the RSE, pages 501–503, 729, Edinburgh, Scotland.
Vizing, V. (1964). On an estimate of the chromatic class of a p-graph. Diskret. Analiz., pages 25–30.
Wang, W. (2002). Equitable total coloring of graphs with maximum degree 3. Graphs Comb, 18:677–685.
Dantas, S., de Figueiredo, C. M. H., Mazzuoccolo, G., Preissman, M., dos Santos, V. F., and Sasaki, D. (2016). On the equitable total chromatic number of cubic graphs. Discrete Appl. Math., 209:84–91.
Gardner, M. (1976). Mathematical games: Snarks, boojums and other conjectures related to the four-color-map theorem. Sci. Am., pages 126–130.
Isaacs, R. (1976). Loupekhine’s snarks: A bi-family of non-tait-colorable graphs. Tec. Report.
M.Behzad (1965). Graphs and Their Chromatic Numbers. PhD thesis, Michigan State University, Michigan.
Sasaki, D., Dantas, S., de Figueiredo, C. M. H., Mazzuoccolo, G., and Preissman, M. (2014). The hunting of a snark with total chromatic number 5. Discrete Appl. Math., 164:470–481.
Tait, P. G. (1878-1880). Remarks on the colouring of maps. In Proceedings of the RSE, pages 501–503, 729, Edinburgh, Scotland.
Vizing, V. (1964). On an estimate of the chromatic class of a p-graph. Diskret. Analiz., pages 25–30.
Wang, W. (2002). Equitable total coloring of graphs with maximum degree 3. Graphs Comb, 18:677–685.
Published
2023-08-06
How to Cite
ARAÚJO, Rieli; SASAKI, Diana.
Coloração total equilibrada dos snarks de Loupekine. In: PROCEEDINGS OF THE THEORY OF COMPUTATION MEETING (ETC), 8. , 2023, João Pessoa/PB.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2023
.
p. 20-24.
ISSN 2595-6116.
DOI: https://doi.org/10.5753/etc.2023.230043.
