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 $c-b\ne b-a$. Consequently, every nice regular graph is universally $3$-edge-weightable.