Decomposing cubic graphs into isomorphic linear forests

TL;DR

This paper proves Wormald's conjecture that large connected cubic graphs with 4n vertices can be decomposed into two isomorphic linear forests, using probabilistic methods and structural analysis.

Contribution

It confirms Wormald's conjecture for large connected cubic graphs, employing novel probabilistic and local recoloring techniques.

Findings

Wormald's conjecture is true for large connected cubic graphs.

The decomposition into isomorphic linear forests is achievable under specified conditions.

The proof introduces new probabilistic and structural methods for graph decomposition.

Abstract

A common problem in graph colouring seeks to decompose the edge set of a given graph into few similar and simple subgraphs, under certain divisibility conditions. In 1987 Wormald conjectured that the edges of every cubic graph on $4n$ vertices can be partitioned into two isomorphic linear forests. We prove this conjecture for large connected cubic graphs. Our proof uses a wide range of probabilistic tools in conjunction with intricate structural analysis, and introduces a variety of local recolouring techniques.