an:05211649
Zbl 1125.05043
Wang, Haiying
On the adjacent vertex-distinguishing total chromatic numbers of the graphs with \(\Delta (G) = 3\)
EN
J. Comb. Optim. 14, No. 1, 87-109 (2007).
00209073
2007
j
05C15
adjacent vertex-distinguishing total coloring; adjacent vertex-distinguishing total chromatic number; subdivision vertex; subdivision graph
Summary: 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 \(f: T(G)\rightarrow C\) such that no adjacent or incident elements of \(T (G)\) receive the same color. For any \(u\in V(G)\), denote \(C(u)=\{f(u)\}\cup\{f(uv)\mid uv\in E(G)\}\). The total \(k\)-coloring \(f\) of \(G\) is called 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.