On decomposing graphs of large minimum degree into locally irregular subgraphs

TL;DR

This paper proves that graphs with sufficiently large minimum degree can be decomposed into three locally irregular subgraphs, advancing the understanding of graph decompositions related to the 1-2-3 Conjecture.

Contribution

It establishes that all large minimum degree graphs, beyond a certain threshold, can be decomposed into three locally irregular subgraphs, confirming a special case of a longstanding conjecture.

Findings

Graphs with minimum degree at least 10^{10} can be decomposed into 3 locally irregular subgraphs.

The result strengthens previous work by Addario-Berry et al.

The approach combines probabilistic methods with degree-constrained subgraph theorems.

Abstract

A \emph{locally irregular graph} is a graph whose adjacent vertices have distinct degrees. We say that a graph $G$ can be decomposed into $k$ locally irregular subgraphs if its edge set may be partitioned into $k$ subsets each of which induces a locally irregular subgraph in $G$. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree $3$, every connected graph can be decomposed into $3$ locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of sufficiently large minimum degree, $\delta(G)\geq 10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].