On the oriented perfect path double cover conjecture

Abstract

An {\sf oriented perfect path double cover} ($\rm OPPDC$) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each edge 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 $\rm OPPDC$ and they proved that the minimum degree of the minimal counterexample to this conjecture is at least four. In this paper, among some other results, we prove that the minimal counterexample to this conjecture is 2-connected and 3-edge-connected.