On the Standard (2,2)-Conjecture

TL;DR

This paper proves the Standard (2,2)-Conjecture for graphs with sufficiently large minimum degree, showing such graphs can be decomposed into subgraphs weighted with just 1 and 2 to distinguish adjacent vertices.

Contribution

It establishes the Standard (2,2)-Conjecture for graphs with minimum degree at least one million, using probabilistic methods and degree-constrained subgraph theorems.

Findings

Graphs with minimum degree ≥ 10^6 can be decomposed into two subgraphs with weights 1 and 2.

The decomposition ensures adjacent vertices have distinct weighted degrees.

The proof employs the Lovász Local Lemma and degree-constrained subgraph theorems.

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.