| Issue |
RAIRO-Oper. Res.
Volume 57, Number 5, September-October 2023
Graphs, Combinatorics, Algorithms and Optimization
|
|
|---|---|---|
| Page(s) | 2619 - 2637 | |
| DOI | https://doi.org/10.1051/ro/2023129 | |
| Published online | 16 October 2023 | |
On total coloring and equitable total coloring of infinite snark families
1
IME, Fluminense Federal University, Niterói, Brazil
2
IME, University of the State of Rio de Janeiro, Rio de Janeiro, Brazil
* Corresponding author: diana.sasaki@ime.uerj.br
Received:
24
January
2023
Accepted:
22
August
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.
Mathematics Subject Classification: 05C15
Key words: Equitable total coloring / total coloring / Blowup snarks / SemiBlowup snarks / superposition
© The authors. Published by EDP Sciences, ROADEF, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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