Small oriented cycle double cover of graphs

A small oriented cycle double cover (SOCDC)} of a bridgeless graph $G$ on $n$ vertices is a collection of at most $n-1$ directed cycles of the symmetric orientation, $G_s$, of $G$ such that each edge of $G_s$ lies in exactly one of the cycles. It is conjectured that every 2-connected graph except two complete graphs $K_4$ and $K_6$ has an $\mathrm{S}\mathrm{O}\mathrm{C}\mathrm{D}\mathrm{C}$. In this paper, we study graphs with $\mathrm{S}\mathrm{O}\mathrm{C}\mathrm{D}\mathrm{C}$ and obtain some properties of the minimal counterexample to this conjecture.