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

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 ≥ 2, can be decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if d / ∈ {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 Lovász Local Lemma.

On the Standard (2,2)-Conjecture

“…(Note that the mentioned random graph G n,p with constant p ∈ (0, 1), which is asymptotically almost surely {1, 2}-weight colourable is very likely "to be close" to a regular graph, and thus often admits uncomplicated and straightforward application of probabilistic tools such as the Lovász Local Lemma -the lack of such a convenience was one of the main obstacles we had to overcome within our random approach.) Thereby we significantly improve the result of [9], which implies in particular that every graph G without isolated edges can be decomposed into 24 {1, 2, 3}-weight colourable subgraphs (i.e. subgraphs fulfilling the 1-2-3 Conjecture), or at most 2 such subgraphs if G is dregular, d ≥ 18.…”