On the Lucky labeling of Graphs

TL;DR

This paper introduces the concept of lucky labelings in graphs, establishes bounds on the lucky number for most graphs, and provides an algorithm for constructing such labelings.

Contribution

It defines lucky labelings, proves bounds on the lucky number for all graphs except K2, and presents an algorithm for lucky labeling.

Findings

Lucky number bounds between rac{w}{n-w+1} and riangle^2.

The bounds apply to all graphs except K2.

An algorithm for constructing lucky labelings is provided.

Abstract

Suppose the vertices of a graph $G$ were labeled arbitrarily by positive integers, and let $Sum(v)$ denote the sum of labels over all neighbors of vertex $v$. A labeling is lucky if the function $Sum$ is a proper coloring of $G$, that is, if we have $Sum(u) \neq Sum(v)$ whenever $u$ and $v$ are adjacent. The least integer $k$ for which a graph $G$ has a lucky labeling from the set $\lbrace 1, 2, ...,k\rbrace$ is the lucky number of $G$, denoted by $\eta(G)$. We will prove, for every graph $G$ other than $ K_{2} $, $\frac{w}{n-w+1}\leq\eta (G) \leq \Delta^{2} $ and we present an algorithm for lucky labeling of $ G $.