Long antipaths and anticycles in oriented graphs

TL;DR

The paper addresses the problem of guaranteeing long antipaths or anticycles in oriented graphs through the minimum pseudo-semi-degree $\tilde{\delta}^{0}(D)$. It introduces a strengthened threshold and proves that for all $k\ge 2$, $\tilde{\delta}^{0}(D)\ge (2k+1)/3$ ensures either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$, improving prior bounds. The authors employ a detailed analysis of longest antipaths and anticycles, including a recursive refinement technique with $f(k)=(2k+1)/3$ and $\alpha=\lceil \log_{2} k\rceil$, to derive a contradiction and establish the result. They also present a construction showing the bound is not improvable in general and address a related open question, providing a negative answer to a problem posed by Klimus̆ová and Stein. The results advance understanding of orientated graph rigidity under degree constraints and clarify the limitations of $\tilde{\delta}^{0}(D)$-based forcing for long anti-oriented structures.

Abstract

Let $δ^{0}(D)$ be the minimum semi-degree of an oriented graph $D$. Jackson (1981) proved that every oriented graph $D$ with $δ^{0}(D)\geq k$ contains a directed path of length $2k$ when $|V(D)|>2k+2$, and a directed Hamilton cycle when $|V(D)|\le 2k+2$. Stein~(2020) further conjectured that every oriented graph $D$ with $δ^{0}(D)>k/2$ contains any orientated path of length $k$. Recently, Klimousová and Stein (DM, 2023) introduced the minimum pseudo-semi-degree $\tildeδ^0(D)$ (a slight weaker than the minimum semi-degree condition as $\tildeδ^0(D)\ge δ^0(D))$ and showed that every oriented graph $D$ with $\tildeδ^{0}(D)\ge (3k-2)/4$ contains each antipath of length $k$ for $k\geq 3$. In this paper, we improve the result of Klimousová and Stein by showing that for all $k\geq 2$, every oriented graph with $\tildeδ^0(D)\ge(2k+1)/3$ contains either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$. Furthermore, we answer a problem raised by Klimousová and Stein in the negative.

Theorems & Definitions (8)

• Conjecture 1.1
• Conjecture 1.4
• Lemma 1.7
• Corollary 1.8
• Lemma 2.2
• proof
• proof
• proof