Supereulerian Oriented Graphs With Large Arc-Strong Connectivity

Abstract

An oriented graph is a digraph whose underlying graph is simple. Bang-Jensen and Thomassé conjectured that every digraph $D$ with arc-strong connectivity $\lambda \left(D\right)$ at least as large as its independence number $\alpha \left(D\right)$ must be supereulerian. We introduce max–min ditrails in a digraph and investigate the relationship between the arc-strong connectivity $\lambda \left(D\right)$ and matching number ${\alpha }^{\prime }\left(D\right)$ to assure the supereulericity of an oriented graph $D$. Utilizing the max–min ditrails with the related counting arguments, it is proved that every oriented graph $D$ with $\lambda \left(D\right)\ge \lfloor \frac{{\alpha }^{\prime }\left(D\right)}{2}\rfloor +1$ is supereulerian. This bound is the best possible.