Long antipaths and anticycles in oriented graphs

Let δ0(D) be the minimum semi-degree of an oriented graph D. Jackson (1981) proved that every oriented graph D with δ0(D)≥k contains a directed path of length 2k when |V(D)|>2k+2, and a directed Hamilton cycle when |V(D)|≤2k+2. Stein (2020) further conjectured that every oriented graph D with δ0(D)>k/2 contains any orientated path of length k. Recently, Klimošová and Stein (2023) introduced the minimum pseudo-semi-degree δ˜0(D) (A slightly weaker variant of the minimum semi-degree condition as δ˜0(D)≥δ0(D)) and showed that every oriented graph D with δ˜0(D)≥(3k−2)/4 contains each antipath of length k for k≥3. In this paper, we improve the result of Klimošová and Stein by showing that for all k≥2, every oriented graph with δ˜0(D)≥(2k+1)/3 contains either an antipath of length at least k+1 or an anticycle of length at least k+1.