Improved Bounds on the Span of L(1,2)-edge Labeling of Some Infinite Regular Grids
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For two given nonnegative integers h and k, an L(h,k)-edge labeling of a graph G is the assignment of labels {0,1, ⋯, n} to the edges so that two edges having a common vertex are labeled with difference at least h and two edges not having any common vertex but having a common edge connecting them are labeled with difference at least k. The span λ'_h,k(G) is the minimum n such that G admits an L(h,k)-edge labeling. Here our main focus is on finding λ'_1,2(G) for L(1,2)-edge labeling of infinite regular hexagonal (T_3), square (T_4), triangular (T_6) and octagonal (T_8) grids. It was known that 7 ≤λ'_1,2(T_3)≤ 8, 10 ≤λ'_1,2(T_4)≤ 11, 16 ≤λ'_1,2(T_6)≤ 20 and 25 ≤λ'_1,2(T_8)≤ 28. Here we settle two long standing open questions i.e. λ'_1,2(T_3) and λ'_1,2(T_4). We show λ'_1,2(T_3) =7, λ'_1,2(T_4)= 11. We also improve the bound for T_6 and T_8 and prove λ'_1,2(T_6)≥ 18, λ'_1,2(T_8)≥ 26.
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