Abstract
A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by \(\chi ''(G)\), is the minimum number of colors in a total coloring of G. The well-known total coloring conjecture (TCC) says that every graph with maximum degree \(\Delta \) admits a total coloring with at most \(\Delta + 2\) colors. A graph is 1-toroidal if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree \(\Delta \) at least 11 and without adjacent triangles, then G admits a total coloring with at most \(\Delta + 2\) colors.
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Acknowledgments
This project was supported by the National Natural Science Foundation of China (11101125) and partially supported by the Fundamental Research Funds for Universities in Henan. The author would like to thank the anonymous reviewers for their valuable comments and assistance on earlier drafts.
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Wang, T. Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles. J Comb Optim 33, 1090–1105 (2017). https://doi.org/10.1007/s10878-016-0025-9
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DOI: https://doi.org/10.1007/s10878-016-0025-9