Path decompositions of random directed graphs
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.