The List Linear Arboricity of Digraphs

TL;DR

This paper extends the bounds on the list linear arboricity of digraphs, showing that for large maximum degree, the minimum number of directed linear forests needed to decompose a digraph is close to its maximum degree, with a logarithmic error term.

Contribution

It generalizes the Lang and Postle bound for undirected graphs to directed graphs, providing a near-optimal upper bound in the list setting for large maximum degree.

Findings

Proves $la(D) \,\leq\, \Delta + 6\sqrt{\Delta} \log^4 \Delta$ for large $\Delta$.

Establishes the same bound in the list coloring setting.

Extends known bounds from undirected to directed graphs.

Abstract

A (directed) linear forest is a (di)graph whose components are (directed) paths. The linear arboricity $la(F)$ of a (di)graph $F$ is the minimum number of (directed) linear forests required to decompose its edges. Akiyama, Exoo, and Harary (1980) proposed the Linear Arboricity Conjecture that $la(G) \leq \left\lceil \frac{\Delta+1}{2}\right\rceil$ for any graph $G$ of maximum degree $\Delta$. The current best known bound, due to Lang and Postle (2023), establishes $la(G) \leq \frac{\Delta}{2} + 3\sqrt{\Delta} \log^4 \Delta$ for sufficiently large $\Delta$. And they proved this in the stronger list setting proposed by An and Wu. For a digraph $D$, let its maximum degree $\Delta(D)$ be the maximum of all in-degrees and out-degrees of its vertices. Nakayama and P\'{e}roche (1987) conjectured that $la(D) \leq \Delta(D)+1$ for every digraph $D$. We extend Lang and Postle's result to digraphs with a matching error term. We show that $la(D) \leq\Delta + 6\sqrt{\Delta} \log^4 \Delta$ for any digraph $D$ with $\Delta = \Delta(D)$ sufficiently large. Moreover, we also establish this bound in the stronger list setting, where each arc $e \in A(D)$ is assigned a list of colors, and each arc is assigned a color from its list such that each color class forms a directed linear forest.