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Article Dans Une Revue Discrete Applied Mathematics Année : 2016

On the equitable total chromatic number of cubic graphs

Résumé

A total coloring is equitable if the number of elements colored with each color differs by at most one, and the least integer for which a graph has such a coloring is called its equitable total chromatic number. Wang conjectured that the equitable total chromatic number of a multigraph of maximum degree Delta is at most Delta + 2, and proved this for the case where Delta <= 3. Therefore, the equitable total chromatic number of a cubic graph is either 4 or 5, and in this work we prove that the problem of deciding whether it is 4 is NP-complete for bipartite cubic graphs. Furthermore, we present the first known Type 1 cubic graphs with equitable total chromatic number 5. All of them have, by construction, a small girth. We also find one infinite family of Type 1 cubic graphs of girth 5 having equitable total chromatic number 4. This motivates the following question: Does there exist Type 1 cubic graphs of girth greater than 5 and equitable total chromatic number 5

Dates et versions

hal-01355517 , version 1 (23-08-2016)

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Simone Dantas, Celina de Figueiredo, Giuseppe Mazzuoccolo, Myriam Preissmann, Vinicius Fernandes Dos Santos, et al.. On the equitable total chromatic number of cubic graphs. Discrete Applied Mathematics, 2016, 9th International Colloquium on Graph Theory and Combinatorics (ICGT) jun 2014 Grenoble, 209 (supl), pp.84 - 91. ⟨10.1016/j.dam.2015.10.013⟩. ⟨hal-01355517⟩
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