A note about a theorem by Maamoun on decompositions of digraphs into elementary directed paths or cycles (Q1066157)

Maamoun has proved that there exists an elementary directed path or an elementary directed cycle which meets every demi-cocycle of maximum size. If \(\alpha\) denotes the maximum size of a demi-cocycle, then there exists a partition of a digraph into \(\alpha\) elementary directed paths or elementary directed cycles. The present paper shows that there is a partition with at most \(\alpha\) elementary directed paths or elementary directed cycles and with exactly \(\sum_{x}\max [d^+_ G(x)-d^-_ G(x),0]\) elementary directed paths, where \(d^+_ G(x)\) and \(d^-_ G(x)\) denote the outdegree and indegree of a vertex x of the digraph G, respectively.