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

Haiying Wang ()
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Haiying Wang: Beijing Institute of Technology

Journal of Combinatorial Optimization, 2007, vol. 14, issue 1, No 6, 87-109

Abstract: 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.

Keywords: The adjacent vertex-distinguishing total coloring; The adjacent vertex-distinguishing total chromatic number; Subdivision vertex; Subdivision graph (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10878-006-9038-0

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