Optimal channel assignment and L(p, 1)-labeling

Abstract

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function $$f:V(G)\rightarrow \{0,1,2,\ldots ,k\}$$ such that $$|f(u)-f(v)|\ge p$$ if $$d(u,v)=1$$ and $$|f(u)-f(v)|\ge 1$$ if $$d(u,v)=2$$ , where d(u, v) is the distance between the two vertices u and v in the graph. Denote $$\lambda _{p,1}^l(G)= \min \{k \mid G$$ has a list k-L(p, 1)-labeling $$\}$$ . In this paper we show upper bounds $$\lambda _{1,1}^l(G)\le \Delta +9$$ and $$\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}$$ for planar graphs G without 4- and 6-cycles, where $$\Delta $$ is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.