$L(p,q)$-labeling and integer tension of a graph embedded on torus

Abstract

The $L(p, q)$-labeling arises from the optimization problem of channel assignment in communication networks. For two non-negative integers $p$ and $q$, an $L(p,q)$-labeling $c$ of a graph $G$ is an assignment of non-negative integers to the vertices of $G$ such that adjacent vertices are labelled using colors at least $p$ apart, and vertices with distance two are labelled using colors at least $q$ apart. In this paper we establish a connection between an $L(p, q)$-labeling and an integer tension of a graph, which extends a corresponding result for planar graphs. This connection provides us with an effective way to design an $L(p, q)$-labeling for non-planar graphs, in particular for graphs embedded on torus, by choosing a proper cycle basis consisting of facial cycles and some specified cycles of the embedded graph. As an application, we use this method to optimize the edge span for the Cartesian product of two cycles.