On the Standard (2,2)-Conjecture

Abstract

The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge 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 graph with minimum degree $\delta\geq 10^6$ can be decomposed into two subgraphs requiring just weights $1$ and $2$ for the same goal. We thus prove the so-called Standard $(2,2)$-Conjecture for graphs with sufficiently large minimum degree. The result is in particular based on applications of the Lov\'asz Local Lemma and theorems on degree-constrained subgraphs.