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Linear arboricity of robust expanders

Yuping Gao, Songling Shan

TL;DR

The paper proves the Linear Arboricity Conjecture for regular robust expanders with linear minimum degree, showing $\mathrm{la}(G)\le \lceil (\Delta(G)+1)/2\rceil$. The authors develop a robust-expander toolkit—covering stability under edits, orientation to robust outexpanders, and a layout-based framework for edge-disjoint spanning configurations—to reduce irregular, near-regular graphs to regularly decomposable hosts and obtain linear-forest decompositions via Hamilton decompositions. They integrate degree-sequence results and matchings in almost-regular graphs to manage irregularity and to facilitate the regularization steps. The results extend the conjecture to dense quasirandom graphs and large graphs with minimum degree close to $n/2$, providing constructive, polynomial-time decompositions and broad applicability to dense graph classes.

Abstract

In 1980, Akiyama, Exoo, and Harary conjectured that any graph $G$ can be decomposed into at most $\lceil(Δ(G)+1)/2\rceil$ linear forests. We confirm the conjecture for robust expanders of linear minimum degree. As a consequence, the conjecture holds for dense quasirandom graphs of linear minimum degree as well as for large $n$-vertex graphs with minimum degree arbitrarily close to $n/2$ from above.

Linear arboricity of robust expanders

TL;DR

The paper proves the Linear Arboricity Conjecture for regular robust expanders with linear minimum degree, showing . The authors develop a robust-expander toolkit—covering stability under edits, orientation to robust outexpanders, and a layout-based framework for edge-disjoint spanning configurations—to reduce irregular, near-regular graphs to regularly decomposable hosts and obtain linear-forest decompositions via Hamilton decompositions. They integrate degree-sequence results and matchings in almost-regular graphs to manage irregularity and to facilitate the regularization steps. The results extend the conjecture to dense quasirandom graphs and large graphs with minimum degree close to , providing constructive, polynomial-time decompositions and broad applicability to dense graph classes.

Abstract

In 1980, Akiyama, Exoo, and Harary conjectured that any graph can be decomposed into at most linear forests. We confirm the conjecture for robust expanders of linear minimum degree. As a consequence, the conjecture holds for dense quasirandom graphs of linear minimum degree as well as for large -vertex graphs with minimum degree arbitrarily close to from above.
Paper Structure (6 sections, 21 theorems, 26 equations)

This paper contains 6 sections, 21 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Degree sequences and number of maximum degree vertices
  4. Matchings in almost-regular graphs
  5. Robust expanders
  6. Proof of Theorem \ref{['thm:expander']}

Key Result

Theorem 1.2

For every $\alpha>0$ there exists $\tau=\tau(\alpha)> 0$ such that for every $\nu> 0$ there exists an integer $N_0=N_0(\alpha, \nu, \tau)$ for which the following holds. Suppose that Then $G$ has a Hamilton decomposition. Moreover, this decomposition can be found in polynomial time in $n$.

Theorems & Definitions (25)

  • Conjecture 1.1: Linear Arboricity Conjecture
  • Theorem 1.2: KO2014
  • Theorem 1.3
  • Theorem 1.4: GKO2016
  • Theorem 1.5
  • Lemma 1.6: GKO2016
  • Lemma 1.7: KO2014
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 2.1: H1962
  • ...and 15 more