On the Span of $l$ Distance Coloring of Infinite Hexagonal Grid

Abstract

For a graph $G(V,E)$ and $l \in \mathbb{N}$, an $l$ distance coloring is a coloring $f: V \to \{1, 2, \cdots, n\}$ of $V$ such that $\forall u,\;v \in V,\; u\neq v,\; f(u)\neq f(v)$ when $d(u,v) \leq l$. Here $d(u,v)$ is the distance between $u$ and $v$ and is equal to the minimum number of edges that connect $u$ and $v$ in $G$. The span of $l$ distance coloring of $G$, $\lambda ^{l}(G)$, is the minimum $n$ among all $l$ distance coloring of $G$. 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 $T_H$, and hence determining $\lambda ^{l}(T_H)$ has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, $2005$] determined the exact value of $\lambda ^{l}(T_H)$ for any odd $l$ and for even $l \geq 8$, it is conjectured that $\lambda ^{l}(T_H) = \left[ \dfrac{3}{8} \left( \, l+\dfrac{4}{3} \right) ^2 \right]$ where $[x]$ is an integer, $x\in \mathbb{R}$ and $x-\dfrac{1}{2} < [x] \leq x+\dfrac{1}{2}$. For $l=8$, the conjecture has been proved by Sasthi and Subhasis [$22$nd Italian Conference on Theoretical Computer Science, $2021$]. In this paper, we prove the conjecture for any $l \geq 10$.