Two-block paths in oriented graphs under the Ore-type condition

Abstract

Stein conjectured that for every positive integer $k$, every oriented graph with minimum semi-degree greater than $k/2$ contains every oriented path of length $k$ [Springer Optim.Appl. 165 (2020), 459--486]. For directed paths this is a classical result of Jackson (1981). The two-block case of the conjecture was established by Chen, Hou and Zhou (2025). In this paper we prove the two-block analogue of Jackson's theorem. For all positive integers $s$ and $r$, let $P(s,r)$ be the oriented path consisting of two directed paths of lengths $s$ and $r$ with a common terminal vertex, and let $Q(s,r)$ be its reverse. We show that for all $K\ge3$ and all $s,r\ge1$ with $s+r\le K$, every oriented graph $D$ with $δ(D)\ge K$ and $d_D^+(u)+d_D^-(v)\ge K-1$ for every ordered pair of distinct vertices $(u,v)\notin A(D)$ contains both $P(s,r)$ and $Q(s,r)$. Our result is the two-block analogue of Jackson's theorem, and it extends the two-block result of Chen, Hou and Zhou. The proof iterates a shift of the terminal vertex of a longest path.