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

TL;DR

This paper proves Stein's conjecture for oriented paths with two blocks in large semi-degree oriented graphs, extending known results from directed paths to a broader class of paths.

Contribution

It establishes the validity of Stein's conjecture specifically for oriented paths with two blocks, a significant extension of previous results.

Findings

Stein's conjecture holds for paths with two blocks in large semi-degree graphs.

The result generalizes Jackson's theorem for directed paths.

The paper advances understanding of path existence in oriented graphs.

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 $P$.