Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

TL;DR

This paper demonstrates that most regular graphs can be decomposed into a small number of subgraphs satisfying the 1-2-3 Conjecture, and improves the general upper bound for such decompositions in arbitrary graphs.

Contribution

It establishes new bounds on decomposing graphs into subgraphs that fulfill the 1-2-3 Conjecture, including specific results for regular graphs and an improved universal bound.

Findings

Most regular graphs (except some degrees) can be decomposed into at most 2 subgraphs fulfilling the 1-2-3 Conjecture.

Any graph without isolated edges can be decomposed into at most 24 such subgraphs, improving previous bounds.

The Lovász Local Lemma is used to achieve these decomposition results.

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. We prove that every $d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if $d\notin\{10,11,12,13,15,17\}$, and into at most $3$ such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most $24$ subgraphs fulfilling the 1-2-3 Conjecture, improving the previously best upper bound of $40$. Both results are partly based on applications of the Lov\'asz Local Lemma.