Path decompositions of oriented graphs

TL;DR

This paper investigates the minimal path decompositions of directed graphs, confirming conjectures for random regular graphs and large-girth graphs, advancing understanding of graph orientations and decompositions.

Contribution

It proves Pullman's conjecture for random odd-regular graphs with high probability and for graphs with sufficiently large girth, extending previous results on graph decompositions.

Findings

Confirmed Pullman's conjecture for random odd-regular graphs asymptotically almost surely.

Verified the conjecture for graphs with sufficiently large girth.

Extended the understanding of path decompositions in various classes of graphs.

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 Gir\~ao, Granet, K\"uhn, 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).