Long antipaths and anticycles in oriented graphs

Abstract

Let ${\delta }^0\left(D\right)$ be the minimum semi-degree of an oriented graph D. Jackson (1981) proved that every oriented graph D with ${\delta }^0\left(D\right)\ge k$ contains a directed path of length 2k when $\left|V\right(D\left)\right|>2k+2$, and a directed Hamilton cycle when $\left|V\right(D\left)\right|\le 2k+2$. Stein (2020) further conjectured that every oriented graph D with ${\delta }^0\left(D\right)>k/2$ contains any orientated path of length k. Recently, Klimošová and Stein (2023) introduced the minimum pseudo-semi-degree ${\tilde{\delta }}^0\left(D\right)$ (A slightly weaker variant of the minimum semi-degree condition as ${\tilde{\delta }}^0\left(D\right)\ge {\delta }^0\left(D\right))$ and showed that every oriented graph D with ${\tilde{\delta }}^0\left(D\right)\ge (3k-2)/4$ contains each antipath of length k for $k\ge 3$. In this paper, we improve the result of Klimošová and Stein by showing that for all $k\ge 2$, every oriented graph with ${\tilde{\delta }}^0\left(D\right)\ge (2k+1)/3$ contains either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$.