Aneulerian digraphs and the determination of those Eulerian digraphs having an odd number of directed Eulerian paths

Abstract

An antidirected path [3] in a digraph is a path with consecutive edges directed either both towards or both away from their common vertex. An aneulerian digraph is a digraph that contains a closed antidirected path passing through each edge once. It is shown that in a 4-valent Eulerian digraph D every two distinct aneulerian subdigraphs are edge disjoint and the set of them cover the edges. A correspondence is given between the aneulerian subdigraphs of D and the 1-difactors. The main theorem states that an Eulerian digraph which has no bivalent vertices has an odd number of directed Eulerian paths iff it is a 4-valent aneulerian digraph.