Two-block paths in oriented graphs of large semidegree

Abstract

We study the existence of oriented paths with two blocks in oriented graphs under semidegree conditions. A block of an oriented path is a maximal directed subpath. Given positive integers $k$ and $\ell$ with $k/2\le \ell < k$, we establish a semidegree function that guarantees the containment of every oriented path with two blocks of sizes $\ell$ and $k-\ell$. As a corollary, we show that every oriented graph with all in- and out-degrees at least $3k/4$ contains every two-block path with $k$ arcs. Our results extend previous work on Stein's conjecture and related problems concerning oriented paths.