On the Span of ℓ Distance Coloring of Infinite Hexagonal Grid

Article Type

Research Article

Publication Title

International Journal of Foundations of Computer Science

Abstract

For a graph G(V, E) and ℓ ∈ N, an ℓ distance coloring is a coloring f : V → {1, 2, . . ., t} of V with t colors such that ∀u, v ∈ V, u ̸= v, f(u) ̸= f(v) when d(u, v) ≤ 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, χℓ(G), is the minimum t 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 H, and hence determining χℓ(H) has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of χℓ(H) for any odd ℓ and for even ℓ ≥ 8, it is conjectured that χℓ(H) = - 3 8 (ℓ + 4 3 )2 - where [x] is an integer, x ∈ R and x− 1 2 < [x] ≤ x+ 1 2 . For ℓ = 8, the conjecture has been proved by Ghosh and Koley [22nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even ℓ ≥ 10.

First Page

791

Last Page

813

DOI

10.1142/S012905412350020X

Publication Date

11-1-2024

Comments

Open Access; Green Open Access

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