Directed Hamilton cycles in digraphs and matching alternating Hamilton cycles in bipartite graphs

TL;DR

This paper improves existing conditions for the existence of Hamilton cycles in digraphs and bipartite graphs by reducing degree bounds, demonstrating that the original results still hold with minimal exceptions.

Contribution

The authors lower the degree bounds in Woodall's and Las Vergnas's theorems by one, maintaining the Hamilton cycle conclusions with a few well-characterized exceptions.

Findings

Lower bounds are tight and cannot be improved further.

Theorems hold with reduced degree conditions except for specific cases.

Results unify and extend previous Hamiltonicity conditions in digraphs and bipartite graphs.

Abstract

In 1972, Woodall raised the following Ore type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w)+d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is contained in a Hamilton cycle. The lower bounds in both results are tight. In this paper, we reduce both bounds by $1$, and prove that the conclusions still hold, with only a few exceptional cases that can be clearly characterized.