On $L(1,2)$-Edge-Labelings of Some Special Classes of Graphs

Abstract:

      For a graph $G$ and two positive integers $j$ and $k$, an $m$-$L(j,k)$-edge-labeling of $G$ is an assignment on the edges to the set $\{0,\ldots,m\}$, such that adjacent edges receive labels differing by at least $j$, and edges which are distance two apart receive labels differing by at least $k$. The $\lambda^{\prime}_{j,k}$-number of $G$ is the minimum $m$ of an $m$-$L(j,k)$-edge-labeling admitted by $G$. In this article, we study the $L(1,2)$-edge-labeling for paths, cycles, complete graphs, complete multipartite graphs, infinite $\Delta$-regular trees and wheels.