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The List Linear Arboricity of Digraphs

Yueping Shi, Ping Hu

TL;DR

This work extends Lang-Postle’s nibble approach from undirected graphs to digraphs to bound the directed list linear arboricity. By modifying the random coloring procedure to produce a partial acyclic (1,1)-edge-coloring and tracking list/neighbor parameters through a Rödl nibble framework, the authors prove la(D) ≤ Δ + 6√Δ log^4 Δ for sufficiently large Δ, and establish the same bound in the list setting. Central to the argument are probabilistic tools (Lovász Local Lemma, Talagrand concentration) and a finishing step using reserved colors to complete the decomposition into directed linear forests. The results extend the Lang-Postle asymptotics to directed graphs, aligning with the conjectured behavior for digraphs and contributing a robust method for list-directed colorings with acyclic constraints.

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 $la(G) \leq \left\lceil \frac{Δ+1}{2}\right\rceil$ for any graph $G$ of maximum degree $Δ$. The current best known bound, due to Lang and Postle (2023), establishes $la(G) \leq \fracΔ{2} + 3\sqrtΔ \log^4 Δ$ for sufficiently large $Δ$. And they proved this in the stronger list setting proposed by An and Wu. For a digraph $D$, let its maximum degree $Δ(D)$ be the maximum of all in-degrees and out-degrees of its vertices. Nakayama and Péroche (1987) conjectured that $la(D) \leq Δ(D)+1$ for every digraph $D$. We extend Lang and Postle's result to digraphs with a matching error term. We show that $la(D) \leqΔ+ 6\sqrtΔ \log^4 Δ$ for any digraph $D$ with $Δ= Δ(D)$ 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.

The List Linear Arboricity of Digraphs

TL;DR

This work extends Lang-Postle’s nibble approach from undirected graphs to digraphs to bound the directed list linear arboricity. By modifying the random coloring procedure to produce a partial acyclic (1,1)-edge-coloring and tracking list/neighbor parameters through a Rödl nibble framework, the authors prove la(D) ≤ Δ + 6√Δ log^4 Δ for sufficiently large Δ, and establish the same bound in the list setting. Central to the argument are probabilistic tools (Lovász Local Lemma, Talagrand concentration) and a finishing step using reserved colors to complete the decomposition into directed linear forests. The results extend the Lang-Postle asymptotics to directed graphs, aligning with the conjectured behavior for digraphs and contributing a robust method for list-directed colorings with acyclic constraints.

Abstract

A (directed) linear forest is a (di)graph whose components are (directed) paths. The linear arboricity of a (di)graph is the minimum number of (directed) linear forests required to decompose its edges. Akiyama, Exoo, and Harary (1980) proposed the Linear Arboricity Conjecture that for any graph of maximum degree . The current best known bound, due to Lang and Postle (2023), establishes for sufficiently large . And they proved this in the stronger list setting proposed by An and Wu. For a digraph , let its maximum degree be the maximum of all in-degrees and out-degrees of its vertices. Nakayama and Péroche (1987) conjectured that for every digraph . We extend Lang and Postle's result to digraphs with a matching error term. We show that for any digraph with sufficiently large. Moreover, we also establish this bound in the stronger list setting, where each arc 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.
Paper Structure (17 sections, 7 theorems, 41 equations)

This paper contains 17 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Modifying the random coloring procedure
  4. Adapting former work
  5. Reservations.
  6. Parameters.
  7. Iterations.
  8. Finishing the proof.
  9. Proof of Claim \ref{['cla:nibbler']}
  10. Proof of Claim \ref{['cla:MR-edge-bounds']}
  11. Expectation
  12. Concentration
  13. Proofs of Claims \ref{['cla:cycle-tweak']} and \ref{['cla:simultaneously']}
  14. Proof of Claim \ref{['cla:cycle-tweak']}
  15. Expectation
  16. ...and 2 more sections

Key Result

Theorem 3

Let $G$ be an undirected graph with maximum degree $\Delta$ sufficiently large, then $lla(G)\leqslant \frac{\Delta}{2} + 3\sqrt{\Delta} \log^4\Delta$.

Theorems & Definitions (35)

  • Conjecture 1: Linear Arboricity Conjecture
  • Conjecture 2: List Linear Arboricity Conjecture
  • Theorem 3: Lang and Postle LP23
  • Theorem 4
  • Definition 5: Dangerous paths
  • Definition 6: Suspicious directed paths
  • Definition 7: color in-neighbors, out-neighbors
  • proof
  • Lemma 9: Lovász Local Lemma
  • Lemma 10: Finishing blow MR13
  • ...and 25 more