On Cycles through Vertices of Large Semidegree in Digraphs

TL;DR

This paper proves that in certain strong digraphs with a cycle of length n-1, there exists a cycle passing through all vertices with large in-degree and out-degree, except for some extremal cases.

Contribution

It establishes a new condition linking near-maximum cycles to large-degree vertices in strong digraphs of odd order.

Findings

Existence of a cycle containing all high-degree vertices under given conditions

Characterization of extremal cases where the cycle does not include all such vertices

Extension of cycle structure understanding in strong digraphs

Abstract

Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some extremal cases).