Abstract
In this work we consider extensions of a conjecture due to Alspach, Mason, and Pullman from 1976. This conjecture concerns edge decompositions of tournaments into as few paths as possible. There is a natural lower bound for the number paths needed in an edge decomposition of a directed graph in terms of its degree sequence; the conjecture in question states that this bound is correct for tournaments of even order. The conjecture was recently resolved for large tournaments, and 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.
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