The 1-2-3 Conjecture almost holds for regular graphs

TL;DR

This paper advances the understanding of the 1-2-3 Conjecture by showing it nearly holds for all regular graphs, requiring only weights 1-4, and proves it for all sufficiently large degree regular graphs.

Contribution

It demonstrates that the 1-2-3 Conjecture holds for regular graphs using weights 1-4 and confirms the conjecture for all large degree regular graphs.

Findings

The 1-2-3 Conjecture is valid for all regular graphs with degree at least 10^8.

Regular graphs can be weighted with {1,2,3,4} to satisfy the conjecture.

The conjecture holds for all regular graphs when the degree is sufficiently large.

Abstract

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We show that for regular graphs it is sufficient to use weights $1$, $2$, $3$, $4$. Moreover, we prove the conjecture to hold for every $d$-regular graph with $d\geq 10^8$.