Linear arboricity of degenerate graphs

TL;DR

This paper proves that for sufficiently high maximum degree, the linear arboricity of any k-degenerate graph equals half its maximum degree, confirming the Linear Arboricity Conjecture for this class.

Contribution

It establishes the exact linear arboricity for k-degenerate graphs with large maximum degree, advancing understanding of the conjecture.

Findings

Linear arboricity equals ceiling of half the maximum degree for large-degree k-degenerate graphs.

Confirms the Linear Arboricity Conjecture for a broad class of graphs with high maximum degree.

Provides a new bound relating degeneracy, maximum degree, and linear arboricity.

Abstract

A linear forest is a union of vertex-disjoint paths, and the linear arboricity of a graph $G$, denoted by $\operatorname{la}(G)$, is the minimum number of linear forests needed to partition the edge set of $G$. Clearly, $\operatorname{la}(G) \ge \lceil\Delta(G)/2\rceil$ for a graph $G$ with maximum degree $\Delta(G)$. On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\operatorname{la}(G) \leq \lceil(\Delta(G)+1) / 2\rceil$ for every graph $ G $. This conjecture has been verified for planar graphs and graphs whose maximum degree is at most $ 6 $, or is equal to $ 8 $ or $ 10 $. Given a positive integer $k$, a graph $G$ is $k$-degenerate if it can be reduced to a trivial graph by successive removal of vertices with degree at most $k$. We prove that for any $k$-degenerate graph $G$, $\operatorname{la}(G) = \lceil\Delta(G)/2 \rceil$ provided $\Delta(G) \ge 2k^2 -k$.