The Linear Arboricity Conjecture for Graphs with Large Girth

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.