On the Span of l Distance Coloring of Infinite Hexagonal Grid
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For a graph G(V,E) and l ∈ℕ, an l distance coloring is a coloring f: V →{1, 2, ⋯, n} of V 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, λ ^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 λ ^l(T_H) has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of λ ^l(T_H) for any odd l and for even l ≥ 8, it is conjectured that λ ^l(T_H) = [ 38( l+43) ^2 ] where [x] is an integer, x∈ℝ and x-12 < [x] ≤ x+12. For l=8, the conjecture has been proved by Sasthi and Subhasis [22nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any l ≥ 10.
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