The Linear Arboricity Conjecture for Graphs with Large Girth

Tapas Kumar Mishra

Paper

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

. For a

-regular graph

, this implies

. In this note, we utilize a network flow construction to establish upper bounds on

conditioned on the girth

. We prove that if

, the conjecture holds true, i.e.,

. Furthermore, we demonstrate that for graphs with girth

at least

and

for any integer constant

, the linear arboricity

satisfies the upper bounds

and

, respectively. Our approach relies on decomposing the graph into

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.