Optimal L(1,2)-edge Labeling of Infinite Octagonal Grid
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For two given non-negative integers h and k, an L(h,k)-edge labeling of a graph G=(V(G),E(G)) is a function f':E(G) {0,1,⋯, n} such that ∀ e_1,e_2 ∈ E(G), | f'(e_1)-f'(e_2) |≥ h when d'(e_1,e_2)=1 and | f'(e_1)-f'(e_2) |≥ k when d'(e_1,e_2)=2 where d'(e_1,e_2) denotes the distance between e_1 and e_2 in G. Here d'(e_1,e_2)=k' if there are at least (k'-1) number of edges in E(G) to connect e_1 and e_2 in G. The objective is to find span which is the minimum n over all such L(h,k)-edge labeling and is denoted as λ'_h,k(G). Motivated by the channel assignment problem in wireless cellular network, L(h,k)-edge labeling problem has been studied in various infinite regular grids. For infinite regular octagonal grid T_8, it was proved that 25 ≤λ'_1,2(T_8) ≤ 28 [Tiziana Calamoneri, International Journal of Foundations of Computer Science, Vol. 26, No. 04, 2015] with a gap between lower and upper bounds. In this paper we fill the gap and prove that λ'_1,2(T_8)= 28.
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