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Two-block paths in oriented graphs of large semidegree

Irena Penev, S Taruni, Stéphan Thomassé, Ana Trujillo-Negrete, Mykhaylo Tyomkyn

TL;DR

The paper addresses the problem of determining semidegree thresholds that guarantee containment of every two-block $k$-arc oriented path with block sizes $\ell$ and $k-\ell$ in an oriented graph. The authors establish a piecewise semidegree function $\delta^0(G)$, proving embeddings of the two-block path via a longest-path strategy and a key structural proposition, and derive the corollary that $\delta^0(G) \ge 3k/4$ suffices to embed all such paths. This work extends Stein's conjecture framework to two-block paths and connects to antidirected-path bounds, enriching the theory of oriented-path embeddings under minimum semidegree constraints. Overall, the results deepen understanding of how local degree conditions enforce global oriented-path structures and provide near-tight bounds with implications for related conjectures in directed graph theory.

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.

Two-block paths in oriented graphs of large semidegree

TL;DR

The paper addresses the problem of determining semidegree thresholds that guarantee containment of every two-block -arc oriented path with block sizes and in an oriented graph. The authors establish a piecewise semidegree function , proving embeddings of the two-block path via a longest-path strategy and a key structural proposition, and derive the corollary that suffices to embed all such paths. This work extends Stein's conjecture framework to two-block paths and connects to antidirected-path bounds, enriching the theory of oriented-path embeddings under minimum semidegree constraints. Overall, the results deepen understanding of how local degree conditions enforce global oriented-path structures and provide near-tight bounds with implications for related conjectures in directed graph theory.

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 and with , we establish a semidegree function that guarantees the containment of every oriented path with two blocks of sizes and . As a corollary, we show that every oriented graph with all in- and out-degrees at least contains every two-block path with arcs. Our results extend previous work on Stein's conjecture and related problems concerning oriented paths.

Paper Structure

This paper contains 5 sections, 4 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary results
  4. Proof of Theorem \ref{['thm:function']}
  5. Concluding remarks

Key Result

Theorem 1.2

Let $k$ and $\ell$ be positive integers with ${k}/{2}\le \ell<k$. Let $G$ be an oriented graph such that Then $G$ contains as a subgraph both possible $k$-arc oriented paths with two blocks of size $\ell$ and $k-\ell$.

Figures (3)

  • Figure 1: The semidegree function given in Theorem \ref{['thm:function']}.
  • Figure 2: Concatenation $\overleftarrow{P_1}P_2$, where $P_1 = xyz$ and $P_2=xuvw$.
  • Figure 3: Examples of paths with two blocks. On the left, $P(\overleftarrow{5},\overrightarrow{4})$ consists of a first block of 5 backward arcs followed by a second block of 4 forward arcs. On the right, $P(\overrightarrow{5},\overleftarrow{5})$ consists of a first block of 5 forward arcs followed by a second block of 5 backward arcs.

Theorems & Definitions (7)

  • Conjecture 1.1: Stein, stein_conjecture
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 3.1: Jackson jackson
  • Proposition 3.2
  • proof
  • proof