Optimal frequency assignment and planar list L(2, 1)-labeling

Abstract

G has a list k-L(2, 1)-labeling if for any k-list assignment L, there exists a coloring $c:V(G)\rightarrow \bigcup \limits _{v\in V} L(v)$ of G such that $c(v)\in L(v)$ for $\forall v\in V(G)$ and for $\forall u,v\in V(G)$, $|c(u)-c(v)|\ge 2$ if $d(u,v)=1$, $|c(u)-c(v)|\ge 1$ if $d(u,v)=2$. $\lambda _{2,1}^{l}(G)=\min \{k|G$ has a list k-L(2, 1)-labeling$\}$ is called the list L(2, 1)-labeling number of G. In this paper, we prove that for planar graph with maximum degree $\Delta \ge 5$, girth $g\ge 13$ and without adjacent 13-cycles, $\lambda _{2,1}^{l}(G)\le \Delta +3$ holds. Moreover, the upper bound $\Delta +3$ is tight.