On the oriented perfect path double cover conjecture

TL;DR

This paper investigates the oriented perfect path double cover conjecture, proving that any minimal counterexample must be 2-connected and 3-edge-connected, advancing understanding of the conjecture's validity.

Contribution

The paper proves that the minimal counterexample to the conjecture is necessarily 2-connected and 3-edge-connected, narrowing the scope of potential counterexamples.

Findings

Minimal counterexamples are 2-connected.

Minimal counterexamples are 3-edge-connected.

Supports the conjecture by restricting counterexample properties.

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.