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Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

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  • Tao Wang

    (Henan University)

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 $$\chi ''(G)$$ χ ′ ′ ( 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 $$\Delta $$ Δ admits a total coloring with at most $$\Delta + 2$$ Δ + 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 $$\Delta $$ Δ at least 11 and without adjacent triangles, then G admits a total coloring with at most $$\Delta + 2$$ Δ + 2 colors.

Suggested Citation

  • Tao Wang, 2017. "Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 1090-1105, April.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0025-9
    DOI: 10.1007/s10878-016-0025-9
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    References listed on IDEAS

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    1. Xin Zhang & Jianfeng Hou & Guizhen Liu, 2015. "On total colorings of 1-planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(1), pages 160-173, July.
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