Universal 3-Edge-Weightability of Regular Graphs

TL;DR

The paper addresses whether every nice graph is universally $3$-edge-weightable, a strengthening beyond the $1$-$2$-$3$ Conjecture. It develops an edge-weighting framework based on independent-set decomposition and a two-phase conflict-resolution scheme to construct proper edge-weightings for any three-element set $\\{a,b,c\\}$ on regular graphs. The main technical result proves that every $k$-regular graph with $k \\ge 3$ admits a proper $\\{-d_1,0,d_2\\}$-edge weighting for any $0<d_1<d_2$, and from this, through shifting and affine considerations, establishes universal $3$-edge-weightability for all triples on regular graphs with $k \\ge 2$. This places regular graphs in a strong position within the hierarchy of edge-weighting properties and suggests broader applicability of the two-phase method to related weighting and labeling problems, while leaving open questions about extending to all nice graphs and to edge-weight choosability. The results underscore the structural leverage provided by regularity for accommodating arbitrary three-element weight sets.

Abstract

A graph $G=(V,E)$ is universally $3$-edge-weightable if for every set $\{a,b,c\}$ of three distinct real numbers, there exists an edge-weighting $w:E\to\{a,b,c\}$ such that $d_w(u)\neq d_w(v)$ for all $uv\in E$. The 1-2-3 Conjecture, recently settled by Keusch, ensures that all nice graphs admit a proper $\{1,2,3\}$-edge-weighting. For regular graphs, this implies proper edge-weightings for arithmetic progressions $\{a,b,c\}$ via affine transformation, but not for general triples due to the spacing-ratio preservation property of affine maps. We prove that every $k$-regular graph with $k\ge 2$ is universally $3$-edge-weightable. This establishes proper edge-weightings for all triples $\{a,b,c\}$, not just arithmetic progressions, showing that regular graphs satisfy a strictly stronger weighting property than what follows from the 1-2-3 Conjecture alone. The proof adapts the independent-set decomposition framework to edge-weightings and introduces a two-phase conflict-resolution scheme that operates independently of the specific choice of $\{a,b,c\}$.

Theorems & Definitions (8)

• Theorem 1
• Corollary 2
• proof
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm1']}