Small oriented cycle double cover of graphs

TL;DR

This paper investigates the existence of small oriented cycle double covers in bridgeless graphs, explores properties of potential minimal counterexamples, and advances understanding of a conjecture related to 2-connected graphs.

Contribution

It studies graphs with small oriented cycle double covers and characterizes properties of minimal counterexamples to the conjecture.

Findings

Identifies properties of minimal counterexamples

Provides partial results supporting the conjecture

Analyzes structural aspects of graphs with SOCDC

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