An edge labeling of graphs from Rados partition regularity condition

TL;DR

This paper introduces the concept of AR-labeling in graphs, where each vertex has distinct subset sum edge weights, and explores the existence of such labelings for different graphs.

Contribution

It formally defines AR-labeling and AR-graphs, initiating the study of their properties and existence conditions in graph theory.

Findings

Defined AR-vertices and AR-labelings in graphs.

Established the concept of AR-graphs and their significance.

Laid groundwork for further research on AR-labelings.

Abstract

A vertex $v$ is called an AR-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. i.e., if $\{x_1,x_2,\dots,x_k\}$ are the edge labels of the edges incident on $v$, then the $2^k$ subset sums are all distinct. An injective edge labeling $f$ of a graph $G$ is said to be an AR-labeling of $G$, if $f:E \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an AR-vertex under $f$. A graph $G$ is said to be an AR-graph, if there exists an AR-labeling $f:E\rightarrow \{1,2,\dots,m\}$, where $m$ denotes the number of edges of $G$. A study of AR-labeling and AR-graphs is initiated in this paper.