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On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3

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  • Haiying Wang

    (Beijing Institute of Technology)

Abstract

Let $$G=(V(G),E(G))$$ be a simple graph and T(G) be the set of vertices and edges of G. Let C be a k-color set. A (proper) total k-coloring f of G is a function $$f\!: T(G)\longrightarrow C$$ such that no adjacent or incident elements of T(G) receive the same color. For any $$u\in V(G)$$ , denote $$C(u)=\{f(u)\}\cup\{f(uv)|uv\in E(G)\}$$ . The total k-coloring f of G is called the adjacent vertex-distinguishing if $$C(u)\neq C(v)$$ for any edge $$uv\in E(G)$$ . And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number $$\chi_{at}(G)$$ of G. In this paper, we prove that $$\chi_{at}(G)\leq 6$$ for all connected graphs with maximum degree three. This is a step towards a conjecture on the adjacent vertex-distinguishing total coloring.

Suggested Citation

  • Haiying Wang, 2007. "On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3," Journal of Combinatorial Optimization, Springer, vol. 14(1), pages 87-109, July.
    • Handle: RePEc:spr:jcomop:v:14:y:2007:i:1:d:10.1007_s10878-006-9038-0
    DOI: 10.1007/s10878-006-9038-0
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    Cited by:

    1. Xiaohan Cheng & Jianliang Wu, 2018. "The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 1-13, January.
    2. Joanna Skowronek-Kaziów, 2017. "Graphs with multiplicative vertex-coloring 2-edge-weightings," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 333-338, January.
    3. Enqiang Zhu & Chanjuan Liu & Jiguo Yu, 2020. "Neighbor product distinguishing total colorings of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 72-76, January.

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