Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Antidirected subgraphs of oriented graphs

Maya Stein, Camila Zárate-Guerén

Abstract

We show that for every $η>0$ every sufficiently large $n$-vertex oriented graph D of minimum semidegree exceeding $(1 + η) k/2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k \ge ηn$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the same setting, D contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length 1 or 2 span a forest. As a special case, we can find all antidirected cycles of length at most $k$. Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

Antidirected subgraphs of oriented graphs

Abstract

We show that for every every sufficiently large -vertex oriented graph D of minimum semidegree exceeding contains every balanced antidirected tree with edges and bounded maximum degree, if . In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the same setting, D contains every -edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length 1 or 2 span a forest. As a special case, we can find all antidirected cycles of length at most . Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in -vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in .
Paper Structure (30 sections, 16 theorems, 34 equations, 3 figures)

This paper contains 30 sections, 16 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof sketches
  3. Connected Antimatchings.
  4. Sketch of the proof of Theorem \ref{['teo:teo2']}
  5. Sketch of the proof of Theorem \ref{['teo:teo3']}
  6. Sketch of the proof of Theorem \ref{['ES']}
  7. Preliminaries
  8. Basic digraph notation
  9. Tree decompositions
  10. Packing trees into edges
  11. Diregularity
  12. Embedding small trees in antiwalks
  13. Connected antimatchings
  14. Antitrees: The proof of Theorem \ref{['teo:teo1']}
  15. Setting the constants.
  16. ...and 15 more sections

Key Result

Theorem 1.2

For all $\eta\in(0,1)$ and $c\in \mathbb N$ there is $n_0$ such that for all $n\geq n_0$ and $k\geq \eta n$, every oriented graph $D$ on $n$ vertices with $\delta^0 (D)>(1 + \eta)\frac{k}{2}$ contains every balanced antidirected tree $T$ with $k$ edges and with $\Delta(T)\leq (\log(n))^c$.

Figures (3)

  • Figure 1: Regularise $D$ to obtain $R$, with a connected antimatching marked in bold.
  • Figure 2: Embedding a small antidirected tree $S$ in $D$.
  • Figure 3: Construction of $D$ from $D'$ in the proof of Lemma \ref{['antilemma']}.

Theorems & Definitions (41)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Conjecture 1.5: Mader mader
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Definition 3.1: $\beta$-decomposition
  • ...and 31 more