The List Linear Arboricity of Graphs

TL;DR

This paper proves that the List Linear Arboricity Conjecture, which extends the classical conjecture on decomposing regular graphs into linear forests, holds true asymptotically, advancing understanding of graph decompositions.

Contribution

The paper establishes that the List Linear Arboricity Conjecture is valid asymptotically, providing a significant extension to the classical linear arboricity results.

Findings

Proves asymptotic validity of the List Linear Arboricity Conjecture.

Extends classical linear arboricity results to list coloring.

Advances theoretical understanding of graph decompositions.

Abstract

A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating that every $d$-regular graph $G$ has linear arboricity $\lceil \frac{d+1}{2} \rceil$. In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC was raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Arboricity Conjecture holds asymptotically.