Path decompositions of oriented graphs
Viresh Patel, Mehmet Akif Yıldız
TL;DR
The paper studies decomposing the edges of a digraph into as few directed paths as possible, focusing on when the path number equals the vertex-excess $ex(D)=\dfrac{1}{2}\sum_v|d^+(v)-d^-(v)|$. It verifies Pullman’s conjecture for random odd $d$-regular graphs and for graphs with sufficiently large girth, introducing a novel sparse absorption technique that replaces robust-expander methods in sparse settings. The authors develop a cycle-absorption framework around plus-minus paths to achieve a perfect path decomposition of size $ex(D)$, even when some vertices have zero excess but are sufficiently separated. These results advance understanding of when orientations of regular (and near-regular) graphs admit minimal path decompositions and provide new tools for handling sparse digraph decomposition problems with probabilistic and structural techniques.
Abstract
We consider the problem of decomposing the edges of a digraph into as few paths as possible. A natural lower bound for the number of paths in any path decomposition of a digraph $D$ is $\frac{1}{2}\sum_{v\in V(D)}|d^+(v)-d^-(v)|$; any digraph that achieves this bound is called consistent. Alspach, Mason, and Pullman conjectured in 1976 that every tournament of even order is consistent and this was recently verified for large tournaments by Girão, Granet, Kühn, Lo, and Osthus. A more general conjecture of Pullman states that for odd $d$, every orientation of a $d$-regular graph is consistent. We prove that the conjecture holds for random $d$-regular graphs with high probability i.e. for fixed odd $d$ and as $n \to \infty$ the conjecture holds for almost all $d$-regular graphs. Along the way, we verify Pullman's conjecture for graphs whose girth is sufficiently large (as a function of the degree).