Path decompositions of digraphs

Abstract

Let G = (X, U) be a digraph of order n. P(G) denotes the minimal cardinal of a path-partition of the arcs of G.Oystein Ore, Theory of graphs (Amer. Math. Soc, Providence, Rhode Island, 1962) has proved that , where . We say that G satisfies Q if the preceeding inequality is an equality.We give some properties of the digraphs satisfying Q, and in particular the case where G is strongly connected. Then we prove that P(G) ≤ [n2/4], and that this result is the best possible. Next we exhibit the existence of digraphs with circuits such that P(G) = [n2/4].Finally we prove that if G is a strongly connected digraph of order n which satisfies Q, then there exists a strongly connected digraph H of order n + 1 having G as a sub-digraph and satisfying Q, too.