Ore-type condition for antidirected Hamilton cycles in oriented graphs

TL;DR

This paper establishes the exact minimum degree condition in large oriented graphs that guarantees the existence of antidirected Hamilton cycles, extending classical results in graph theory.

Contribution

It provides the precise Ore-type degree threshold for antidirected Hamilton cycles in large oriented graphs, improving and generalizing previous bounds.

Findings

The degree threshold for antidirected Hamilton cycles is (3n+2)/4 for large even n.

The proven threshold is shown to be optimal.

The result extends classical theorems to a new class of cycles in oriented graphs.

Abstract

An antidirected cycle in a digraph $G$ is a subdigraph whose underlying graph is a cycle, and in which no two consecutive edges form a directed path in $G$. Let $\sigma_{+-}(G)$ be the minimum value of $d^+(x)+d^-(y)$ over all pairs of vertices $x, y$ such that there is no edge from $x$ to $y$, that is, $$\sigma_{+-}(G)=\min\{d^+(x)+d^-(y): \{x,y\}\subseteq V(G), xy\notin E(G)\}.$$ In 1972, Woodall extended Ore's theorem to digraphs by showing that every digraph $G$ on $n$ vertices with $\sigma_{+-}(G)\geqslant n$ contains a directed Hamilton cycle. Very recently, this result was generalized to oriented graphs under the condition $\sigma_{+-}(G)\geqslant(3n-3)/4$. In this paper, we give the exact Ore-type degree threshold for the existence of antidirected Hamilton cycles in oriented graphs. More precisely, we prove that for sufficiently large even integer $n$, every oriented graph $G$ on $n$ vertices with $\sigma_{+-}(G)\geqslant(3n+2)/4$ contains an antidirected Hamilton cycle. Moreover, we show that this degree condition is best possible.