Erdős Conjecture and AR-Labeling

TL;DR

The paper investigates AR-labelings, where an AR-vertex has distinct subset-sum edge weights for all incident edge subsets, and an AR-graph admits an edge labeling using $\{1,\dots,m\}$ with every vertex AR. Central to the analysis is the ES-sequence from Erdős's subset-sum conjecture, which provides bounds on the AR-index $ARI(G)$ via $ES(\Delta(G)) \le ARI(G) \le ES(m(G))$. These bounds are applied to classify AR-graphs in several graph families, yielding exact AR-index values for stars, wheels, and small complete and bipartite graphs, while showing finiteness results for AR-graphs in certain classes (e.g., bistars, complete graphs, and complete bipartite graphs). The work also discusses the need for more ES-sequence data and computational approaches, and suggests avenues such as Ramsey-type results for Erdős subset-sum questions. Overall, the paper links combinatorial labeling with subset-sum theory to advance understanding of which graphs admit AR-labelings and how large the labeling set must be.

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 Erdős 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.

Figures (6)

• Figure 1: $AR$-Labeling of $B_{2,2}$ and $B_{3,3}$
• Figure 2: $AR$-Labeling of $K_4$ and $K_5$
• Figure 3: $AR$-Labeling of complete bipartite graphs
• Figure 4: $AR$-Labeling of Multipartite graphs $K_{2,2,2}$ and $K_{2,2,3}$
• Figure 5: $AR$-Labeling of $W_6$ and $W_7$

Theorems & Definitions (21)

• Definition : $AR$-Index of G
• Theorem 1
• proof
• Corollary
• proof
• Theorem 2
• proof
• Corollary
• Theorem 3
• proof