Regular graphs are universally 3-edge-weightable

TL;DR

The paper proves that all nice regular graphs can be properly edge-weighted with any 3-element set of real numbers, extending previous results related to the 1-2-3 conjecture.

Contribution

It establishes that all nice regular graphs are universally 3-edge-weightable for any 3-element set with non-arithmetic progression differences.

Findings

Every nice regular graph admits a proper edge weighting from any 3-element set with non-arithmetic progression.

Extends the 1-2-3 conjecture to all 3-element sets with specific difference conditions.

Confirms that nice regular graphs are universally 3-edge-weightable.

Abstract

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular graph has a proper $\{a,b,c\}$-edge weighting. We prove that this remains valid for all 3-element set $\{a,b,c\}$ with $c-b \neq b-a$. Consequently, every nice regular graph is universally $3$-edge-weightable.