An edge labeling of graphs from Rados partition regularity condition
Arun J Manattu, Aparna Lakshmanan S
TL;DR
This work introduces AR-labeling, a graph edge-labeling paradigm inspired by partition regularity, where each vertex must exhibit unique subset-sum behavior on its incident edge labels. It establishes fundamental AR-vertex properties, proves several infinite AR-graph families, and provides explicit constructions for complex structures (Sierpiński graphs, perfect binary/ternary trees, and their glued variants), often leveraging k-shift and subset-sum lemmas. A key contribution is the identification of broad AR-graph classes (including all Hamiltonian cubic graphs) and the explicit labeling schemes that certify AR-graphs, aided by the AR-index as a measure of labeling efficiency. The paper concludes with open problems (notably whether all cubic graphs are AR-graphs) and a general result that every graph admits some AR-labeling, prompting future exploration of AR-index bounds and structural classifications.
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.