Optimal L(j,k)-Edge-Labeling of Regular Grids

Abstract

Given a graph [Formula: see text] and two positive integers j and k, an [Formula: see text]-edge-labeling is a function f assigning to edges of E colors from a set [Formula: see text] such that [Formula: see text] if e and e′ are adjacent, i.e. they share a common endpoint, [Formula: see text] if e and e′ are not adjacent and there exists an edge adjacent to both e and e′. The aim of the [Formula: see text]-edge-labeling problem consists of finding a coloring function f such that the value of [Formula: see text] is minimum. This minimum value is called [Formula: see text]. This problem has already been studied on hexagonal, squared and triangular grids, but mostly not coinciding upper and lower bounds on [Formula: see text] have been proposed. In this paper we close some of these gaps or find better bounds on [Formula: see text] in the special cases [Formula: see text] and [Formula: see text]. Moreover, we propose tight [Formula: see text]-edge-labelings for eight-regular grids.