Small Oriented Cycle Double Cover of Graphs

Abstract

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 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 \({\mathrm{SOCDC}}\). In this paper, we study graphs with \({\mathrm{SOCDC}}\) and obtain some properties of the minimal counterexample to this conjecture.