ON THE ORIENTED PERFECT PATH DOUBLE COVER CONJECTURE

An oriented perfect path double cover (OPPDC) of a graph G is a collection of irected paths in the symmetric orientation Gs of G such that each arc of Gs lies in exactly one of the paths and each vertex of G appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that every graph except two complete graphs K3 and K5 has an OPPDC and they claimed that the minimum degree of the minimal counterexample to this conjecture is at least four. In the proof of their claim, when a graph is smaller than the minimal counterexample, they missed to consider the special cases K3 and K5 . In this paper, among some other results, we present the complete proof for this fact. Moreover, we prove that the minimal counterexample to this conjecture is 2 -connected and 3 -edge-connected.