Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X41120150201On the oriented perfect path double cover conjecture189200597ENB.Bagheri Gh.Isfahan University of TechnologyB.OmoomiJournal Article20121215An oriented perfect path double cover (OPPDC) of a <br />graph $G$ is a collection of directed paths in the symmetric <br />orientation $G_s$ of <br />$G$ such that <br />each arc <br />of $G_s$ lies in exactly one of the paths and each <br />vertex of $G$ appears just once as a beginning and just once as an <br />end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete <br />Math. 276 (2004) 287-294) conjectured that every graph except <br />two complete graphs $K_3$ and $K_5$ has an OPPDC and they <br />claimed that the minimum degree of the minimal counterexample to <br />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 <br />other results, we present the complete proof for this fact. Moreover, we prove that the minimal counterexample to this <br />conjecture is $2$-connected and $3$-edge-connected.http://bims.iranjournals.ir/article_597_a931117b555271ee48028056b0d0101b.pdf