Ore-type condition for antidirected Hamilton cycles in oriented graphs

TL;DR

This paper establishes an exact Ore-type degree threshold for antidirected Hamilton cycles in large even-order oriented graphs: if $\sigma_{+-}(G)\ge (3n+2)/4$, then $G$ contains an antidirected Hamilton cycle. The authors develop a two-pronged proof strategy, handling robust outexpanders via existing Hamiltonicity results and non-expanders via a delicate partition into $A,B,C,D$, two disjoint special edges, and absorption of a long antidirected path to assemble a Hamilton cycle. They also prove sharpness by constructing infinite families with $\sigma_{+-}(G)=\lceil (3n+2)/4 \rceil - 1$ that do not admit antidirected Hamilton cycles. Overall, the work advances the Ore-type theory for oriented graphs, providing a precise threshold and expanding the toolkit (nice partitions, special edges, and absorption) for antidirected cycle constructions with potential broader implications for all orientations of Hamilton cycles.

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 $σ_{+-}(G)$ 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, $$σ_{+-}(G)=\min\{d^+(x)+d^-(y): \{x,y\}\subseteq V(G), xy\notin E(G)\}.$$ In 1972, Woodall extended Ore's theorem to digraphs by showing that every digraph $G$ on $n$ vertices with $σ_{+-}(G)\geqslant n$ contains a directed Hamilton cycle. Very recently, this result was generalized to oriented graphs under the condition $σ_{+-}(G)\geqslant(3n-3)/4$. 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 $σ_{+-}(G)\geqslant(3n+2)/4$ contains an antidirected Hamilton cycle. Moreover, we show that this degree condition is best possible.

Figures (2)

• Figure 1: An illustration of how to find the desired path $P$. The white diamonds and black circles indicate the vertices in $\mathcal{B}_A\cup \mathcal{B}_C$ and $\delta$-good vertices of $G$, respectively. The red dashed edges are obtained by finding common neighbors due to (XY), (CD), (DD) and (AD). For simplicity, the set $C$ is omitted and $P_1,P_2,P^{\ast}$ are placed outside of the partition $(A,B,C,D)$.
• Figure 2: The oriented graphs in Proposition \ref{['PROP-degreesharp']} with order $8s+6,4s$ and $4s+2$, respectively. The size of each sets is given in Table \ref{['TAB-degreesharp']}. The bold edges indicate that all possible edges are present and have the directed shown. Each of $A$ and $C$ spans an almost regular tournament, that is, the in-degree and out-degree of every vertex differ by at most one. Both $B$ and $D$ are empty sets and, in (a) the oriented graph induced by $B$ and $D$ is an almost regular bipartite tournament. In (b), $A$ has order one and the vertex in $A$ has exactly one in-neighbor and out-neighbor in $C$.

Theorems & Definitions (42)

• Theorem 1.1: woodallPLMS24
• Theorem 1.2: changarXiv2025
• Theorem 1.3
• Proposition 1.4
• Lemma 3.1: debiasioEJC22
• Proposition 3.2
• Definition 1
• Lemma 3.3: kuhnAM237
• Theorem 3.4: taylor2013
• Theorem 3.5: debiasioarXiv2025