Channel Assignment with r-Dynamic Coloring

Abstract

The channel assignment is an important problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh. The r-dynamic coloring is a generalization of the L(1, 1)-labeling. An r-dynamic k-coloring of a graph G is a proper k-coloring such that every vertex v is adjacent to at least min$\{d(v),r\}$ different colors. Denote $\chi _{r}(G)=min\{k \mid G$ has an r-dynamic k-coloring} and $ch_{r}(G)=min\{k \mid G$ has a list r-dynamic k-coloring}. In this paper, we show upper bounds $ch_{r}(G)\le r+5$ for planar graphs G with $g(G)\ge 5$ and $r\ge 15$, $ch_{r}(G)\le r+10$ for graphs G with $mad(G)<\frac{10}{3}$.

Supported by NSFC 11771403.