on the oriented perfect path double cover conjecture

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ 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 $k_3$ and $k_5$ 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 $k_3$ and $k_5$‎. ‎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.