Towards the linear arboricity conjecture

TL;DR

This paper advances the understanding of the linear arboricity conjecture by providing improved bounds for all graphs and better bounds for spectral expanders, along with probabilistic algorithms for decomposition.

Contribution

It proves a tighter upper bound on linear arboricity for all graphs and introduces improved bounds for spectral expanders, along with efficient algorithms.

Findings

Established a bound of la(G) ≤ Δ/2 + O(Δ^{2/3-α}) for all graphs.

Provided better bounds for spectral expander graphs.

Developed probabilistic polynomial-time algorithms for linear forest decompositions.

Abstract

The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous conjecture due to Akiyama, Exoo, and Harary from 1980 asserts that $\text{la}(G)\leq \lceil (\Delta(G)+1)/2 \rceil$, where $\Delta(G)$ denotes the maximum degree of $G$. This conjectured upper bound would be best possible, as is easily seen by taking $G$ to be a regular graph. In this paper, we show that for every graph $G$, $\text{la}(G)\leq \frac{\Delta}{2}+O(\Delta^{2/3-\alpha})$ for some $\alpha > 0$, thereby improving the previously best known bound due to Alon and Spencer from 1992. For graphs which are sufficiently good spectral expanders, we give even better bounds. Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests.