Improved Bounds on the Span of L(1, 2)-edge Labeling of Some Infinite Regular Grids

Abstract

For two given non-negative 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 \(\lambda ^{\prime }_{h,k}{(G)}\) is the minimum n such that G admits an L(h, k)-edge labeling. Here our main focus is on finding \(\lambda ^{\prime }_{h,k}{(G)}\) for L(1, 2)-edge labeling of infinite regular hexagonal (T3), square (T4) and triangular (T6) grids. It was known that \(7 \leq \lambda ^{\prime }_{h,k}{(T_3)} \leq 8\), \(10 \leq \lambda ^{\prime }_{h,k}{(T_4)} \leq 11\) and \(16 \leq \lambda ^{\prime }_{h,k}{(T_6)} \leq 20\). Here we have shown that \(\lambda ^{\prime }_{h,k}{(T_3)} \leq 7\), \(\lambda ^{\prime }_{h,k}{(T_4)} \geq 11\) and \(\lambda ^{\prime }_{h,k}{(T_6)} \geq 19\).