The Linear Arboricity Conjecture for Graphs with Large Girth

TL;DR

This paper proves the Linear Arboricity Conjecture for graphs with large girth using a network flow approach, providing upper bounds on linear arboricity based on girth and degree conditions.

Contribution

It introduces a flow-based method to establish upper bounds on linear arboricity for graphs with specified girth, confirming the conjecture in certain cases and extending known bounds.

Findings

Proves the conjecture for graphs with girth at least 2k.

Provides upper bounds for linear arboricity based on girth and degree.

Uses a novel flow network decomposition approach.

Abstract

The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow construction to establish upper bounds on $la(G)$ conditioned on the girth $g(G)$. We prove that if $g(G) \ge 2k$, the conjecture holds true, i.e., $la(G) \le k+1$. Furthermore, we demonstrate that for graphs with girth $g(G)$ at least $k$, $k/2$, $k/4$ and $2k/c$ for any integer constant $c$, the linear arboricity $la(G)$ satisfies the upper bounds $k+2$, $k+3$, $k+5$ and $k+\left\lceil \frac{3c+2}{2}\right\rceil$, respectively. Our approach relies on decomposing the graph into $k$ edge-disjoint 2-factors and constructing an auxiliary flow network with lower bound constraints to identify a sparse transversal subgraph that intersects every cycle in the decomposition.