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

TL;DR

This paper addresses Stein's conjecture for oriented graphs with minimum semi-degree $\delta^{0}(D) > k/2$, proving that every two-block oriented path of length $k$ (i.e., $P(s,t)$ and $Q(s,t)$ with $s+t=k$) is contained in such digraphs. The authors employ a longest directed path technique, coupled with a detailed partitioned-structure analysis and casework, to rule out counterexamples. Consequently, the conjecture holds for all oriented paths with two blocks, extending the prior one-block case proven by Jackson. The result highlights a dichotomy that two-block paths obey the same semi-degree threshold as single-block paths and motivates exploration toward three-or-more-block paths.

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 $P$.

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Theorems & Definitions (27)

• Theorem 1: J
• Theorem 2: Z
• Conjecture 3
• Theorem 4
• Theorem 5
• Claim 7
• proof
• Claim 8
• proof
• Claim 9