A Chvátal--Erdős type condition for supereulerian digraphs with ${\alpha }_2=4$

Abstract

A digraph is \textbf{supereulerian} if it contains a spanning closed trail. Let ${\alpha }_2(D)$ denote the maximum cardinality of a vertex set inducing no 2-cycle. In this paper, we characterize supereulerianity in a strong digraph $D$ with ${\alpha }_2(D)=4$ by proving that a strong digraph $D$ with ${\alpha }_2(D)=4$ and $\lambda (D)\ge 2$ is supereulerian if and only if $D$ does not belong to an exceptional family $\mathcal{H}$ of $2$-arc-strong digraphs with ${\alpha }_2(D)=4$. Furthermore, every strong digraph satisfying ${\alpha }_2(D)=4$ and $\lambda (D)\ge 3$ is supereulerian.

12 pages, 5 figures