Lucky Edge Labeling of $K_n$ and Special Types of Graphs

Abstract

Let G be a simple graph with vertex set V(G) and edge set E(G) respectively. Vertex set V(G) is labeled arbitrary by positive integers and E(e) denote the edge label such that it is the sum of labels of vertices incident with edge e. The labeling is said to be lucky edge labeling if the edge E(G) is a proper coloring of G, that is, if we have $E(e_{1})\neq E(e_{2})$ whenever $e_{1}$ and $e_{2}$ are adjacent edges. The least integer k for which a graph G has a lucky edge labeling from the set $\{1, 2,\dots, k\}$ is the lucky number of G denoted by $\eta(G)$. A graph which admits lucky edge labeling is the lucky edge labeled graph. In this paper, we proved that complete graph $K_{n}$, tadpole graph $T_{m, n}$ and rectangular book $B_{p}^{4}$ are lucky edge labeled graphs.