Path decompositions of random directed graphs

TL;DR

This paper investigates the edge decomposition of directed graphs into the minimum number of paths, confirming a longstanding conjecture for most random directed graphs using probabilistic methods.

Contribution

It proves that the Alspach, Mason, and Pullman conjecture holds with high probability for random directed graphs, extending its validity beyond tournaments.

Findings

The conjecture is true for most directed graphs.

The conjecture holds with high probability for random directed graphs.

Techniques involve absorption and flow methods.

Abstract

We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree sequence: this is given by the excess of $D$, which is the sum of $|d^+(v) - d^-(v)|/2$ over all vertices $v$ of $D$ (here $d^+(v)$ and $d^-(v)$ are, respectively, the out- and indegree of $v$). A conjecture due to Alspach, Mason and Pullman from 1976 states that this bound is correct for tournaments of even order. The conjecture was recently resolved for large tournaments. Here we investigate to what extent the conjecture holds for directed graphs in general. In particular, we prove that the conjecture holds with high probability for the random directed graph $D_{n,p}$ for a large range of $p$ (thus proving that it holds for most directed graphs). To be more precise, we define a deterministic class of directed graphs for which we can show the conjecture holds, and later show that the random digraph belongs to this class with high probability. Our techniques involve absorption and flows.