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Semidegree threshold for spanning trees in oriented graphs

Pedro Araújo, Giovanne Santos, Maya Stein

Abstract

We show that for all $γ> 0$ and $Δ\in \mathbb{N}$, there is some $n_0$ such that, if $n \geq n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8 + γ)n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $Δ$. This is asymptotically best possible.

Semidegree threshold for spanning trees in oriented graphs

Abstract

We show that for all and , there is some such that, if , then every oriented graph on vertices with minimum semidegree at least contains a copy of each oriented tree on vertices with maximum degree at most . This is asymptotically best possible.
Paper Structure (29 sections, 25 theorems, 80 equations, 3 figures)

This paper contains 29 sections, 25 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Robust expansion
  3. Organisation
  4. Overview
  5. Proof for almost spanning trees
  6. Proof for spanning trees
  7. Using the Blow-Up Lemma
  8. Incorporating the exceptional vertices
  9. Obtaining a perfectly balanced assignment
  10. Preliminaries
  11. On sequences of numbers
  12. On probability and concentration inequalities
  13. On multigraphs
  14. On digraphs
  15. On oriented trees
  16. ...and 14 more sections

Key Result

Theorem 1.1

For every $\gamma > 0$ and $\Delta \in \mathbb N$, there exists $n_0 = n_0(\gamma, \Delta)$ such that the following holds for every $n \geq n_0$. If $G$ is an oriented graph on $n$ vertices with ${\delta^{0}(G) \geq (3/8 + \gamma)n}$, and $T$ is an oriented tree on $n$ vertices with $\Delta(T) \leq

Figures (3)

  • Figure 1: An example of a $(v_i,v_j)$-skewed-traverse of length 5.
  • Figure 2: Incorporation of $u$ when $T$ is bare.
  • Figure 3: Balancing a switchy tree.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.3: hamiltonian-degree
  • Theorem 1.4
  • Lemma 1.5: ko
  • Proposition 3.1
  • Lemma 3.2: Chernoff's bound
  • Lemma 3.3: Janson's inequality
  • Lemma 3.4: Azuma's inequality
  • Theorem 3.5: Vizing's Theorem
  • Lemma 3.6: leafy-bare
  • ...and 25 more