Long antipaths and anticycles in oriented graphs

Abstract

Let $\delta^{0}(D)$ be the minimum semi-degree of an oriented graph $D$. Jackson (1981) proved that every oriented graph $D$ with $\delta^{0}(D)\geq k$ 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 $\delta^{0}(D)>k/2$ contains any orientated path of length $k$. Recently, Klimo\u{s}ov\'{a} and Stein (DM, 2023) introduced the minimum pseudo-semi-degree $\tilde\delta^0(D)$ (a slight weaker than the minimum semi-degree condition as $\tilde\delta^0(D)\ge \delta^0(D))$ and showed that every oriented graph $D$ with $\tilde\delta^{0}(D)\ge (3k-2)/4$ contains each antipath of length $k$ for $k\geq 3$. In this paper, we improve the result of Klimo\u{s}ov\'{a} and Stein by showing that for all $k\geq 2$, every oriented graph with $\tilde\delta^0(D)\ge(2k+1)/3$ 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 Klimo\u{s}ov\'{a} and Stein in the negative.