Paths with two blocks in oriented graphs of large minimum semi-degree

Abstract

Stein (2020) conjectured that for any positive integer $k$, every oriented graph of minimum semi-degree greater than $k/2$ contains every oriented path of length $k$. This conjecture is true for directed paths by a result from Jackson (JGT, 1981). In this paper, we establish the validity of Stein's conjecture specifically for any oriented path with two blocks, where, a block of an oriented path $P$ refers to a maximal directed subpath within $P$.