Group Chromatic Number of Halin Graphs

Abstract

Let $G$ be a graph with a fixed orientation and $A$ an abelian group. Denote by $F(G, A)$ the set of all functions $f: E(G)\rightarrow A$. The graph $G$ is $A$-colorable if for any function $f\in F(G, A)$, there is a map $c: V(G)\rightarrow A$ such that for each directed edge $e=uv$ from $u$ to $v$, $c(u)-c(v)\not = f(uv)$. The group chromatic number $\chi _g(G)$ is the minimum positive integer $m$ such that for any abelian group $A$ with $|A|\ge m$, $G$ is $A$-colorable. In this paper, we prove that for any Halin graph $G$, $\chi _g(G)\le 4$ with equality if and only if $G$ is isomorphic to an odd wheel; for any Pseudo-Halin graph $G$, $\chi _g(G)\le 4$ with equality if and only if $G$ contains a specified subtree.