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Antipaths in oriented graphs

Tereza Klimošová, Maya Stein

Abstract

We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices, and as a consequence, we confirm a weakened antidirected path version of a conjecture of Addario-Berry, Havet, Linhares Sales, Thomassé and Reed.

Antipaths in oriented graphs

Abstract

We show that for any natural number , any oriented graph of minimum semidegree at least contains an antidirected path of length . In fact, a slightly weaker condition on the semidegree sequence of suffices, and as a consequence, we confirm a weakened antidirected path version of a conjecture of Addario-Berry, Havet, Linhares Sales, Thomassé and Reed.
Paper Structure (9 sections, 7 theorems, 12 equations)

This paper contains 9 sections, 7 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of Theorem \ref{['antii']}
  4. Proof of Theorem \ref{['antiii']}
  5. Final remarks and open problems
  6. A lower bound in Theorem \ref{['antii']}.
  7. Other orientations of the path.
  8. Antitrees.
  9. Digraphs.

Key Result

Theorem 2

Let $k\in\mathbb N$ with $k\ge 3$ and let $D$ be an oriented graph with $\bar{\delta}^0(D)\ge (3k-2)/4$. Then $D$ contains each antidirected path of length $k$.

Theorems & Definitions (16)

  • Conjecture 1
  • Theorem 2
  • Conjecture 3: Addario-Berry et al. ABHLSTR
  • Theorem 4
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 6 more