On the equitable total chromatic number of cubic graphs
Discrete Applied Mathematics, ISSN: 0166-218X, Vol: 209, Page: 84-91
2016
- 9Citations
- 5Captures
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Article Description
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 Δ is at most Δ+2, and proved this for the case where Δ≤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?
Bibliographic Details
http://www.sciencedirect.com/science/article/pii/S0166218X15005004; http://dx.doi.org/10.1016/j.dam.2015.10.013; http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=84964506140&origin=inward; https://linkinghub.elsevier.com/retrieve/pii/S0166218X15005004; https://dul.usage.elsevier.com/doi/; https://api.elsevier.com/content/article/PII:S0166218X15005004?httpAccept=text/plain; https://api.elsevier.com/content/article/PII:S0166218X15005004?httpAccept=text/xml; https://dx.doi.org/10.1016/j.dam.2015.10.013
Elsevier BV
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