Erd\H{o}s Conjecture and AR-Labeling

TL;DR

This paper investigates the AR-index of graphs, establishing lower bounds based on Erdős conjecture, and determines the AR-index for specific graph classes like stars and wheels.

Contribution

It introduces a new lower bound for the AR-index derived from Erdős's subset sum conjecture and characterizes AR-graphs among certain graph families.

Findings

Finiteness of AR-graphs among bistars, complete graphs, and bipartite graphs.

Exact AR-index values for stars and wheels.

Lower bounds for AR-index based on Erdős subset sum conjecture.

Abstract

Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an $AR$-labeling of $G$, if $f:E(G) \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an $AR$-vertex under $f$. The minimum $k$ such that there exists an $AR$-labeling $f:E\rightarrow \{1,2,3,\dots,k\}$ is called the $AR$-index of G, denoted by $ARI(G)$. In this paper, using a sequence originating from Erd\H{o}s subset sum conjecture, a lower bound has been obtained for the $AR$-index of a graph and this bound is used to prove that only finitely many bistars, complete graphs and complete bipartite graphs are $AR$-graphs. The exact values of $AR$-index is obtained for stars and wheels.