Degree-balanced decompositions of cubic graphs

TL;DR

This paper proves that most cubic graphs have a spanning subgraph with degrees close to a quarter of the vertices, confirming a recent conjecture for this class of graphs.

Contribution

It establishes the existence of degree-balanced spanning subgraphs in cubic graphs, resolving a conjecture by Alon and Wei.

Findings

Most cubic graphs contain a degree-balanced spanning subgraph.

The result holds with at most three small exceptions.

It confirms a conjecture for the class of cubic graphs.

Abstract

We show that every cubic graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree deviates from $\frac{n}{4}$ by at most $\frac{1}{2}$, up to three exceptions. This resolves the conjecture of Alon and Wei (Irregular subgraphs, Combin. Probab. Comput. 32(2) (2023), 269--283) for cubic graphs.