Small Oriented Cycle Double Cover of Graphs

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 arc 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 SOCDC . In this paper, we study graphs with SOCDC and obtain some properties of the minimal counterexample to this conjecture.