On the Span of ℓ Distance Coloring of Infinite Hexagonal Grid

Abstract

For a graph [Formula: see text] and [Formula: see text], an [Formula: see text] distance coloring is a coloring [Formula: see text] of [Formula: see text] with [Formula: see text] colors such that [Formula: see text] when [Formula: see text]. Here [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] and is equal to the minimum number of edges that connect [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of [Formula: see text] distance coloring of [Formula: see text], [Formula: see text], is the minimum [Formula: see text] among all [Formula: see text] distance coloring of [Formula: see text]. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid [Formula: see text], and hence determining [Formula: see text] has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of [Formula: see text] for any odd [Formula: see text] and for even [Formula: see text], it is conjectured that [Formula: see text] where [Formula: see text] is an integer, [Formula: see text] and [Formula: see text]. For [Formula: see text], the conjecture has been proved by Ghosh and Koley [[Formula: see text]nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even [Formula: see text].