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Embedding trees using minimum and maximum degree conditions

Alexey Pokrovskiy, Leo Versteegen, Ella Williams

TL;DR

The paper proves that for large k and trees T with bounded maximum degree Δ, any host graph G with δ(G)≥⌊2k/3⌋ and Δ(G)≥k contains every such T, extending to multiple degree configurations and asymptotically confirming several Besomi–Pavez-Signé–Stein conjectures. The authors develop a sparse-to-dense strategy via a hyperstability-based decomposition into rich subgraphs, then apply regularity- and density-embedded arguments to finish embeddings in a dense subgraph. They obtain exact 2/3 and α-exact results for bounded-degree trees, along with second-neighbour embedding results, and provide a coherent framework for extending sparse embedding problems to dense settings. The work advances the understanding of mixed-degree conditions in tree embeddings and lays groundwork for further exact and asymptotic results in sparse regimes.

Abstract

A variant of the Erdős-Sós conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges. Both degree bounds are best possible. We confirm this conjecture for large trees with bounded maximum degree, by proving that for all $Δ\in \mathbb{N}$ and sufficiently large $k\in \mathbb{N}$, every graph $G$ with $δ(G)\geq \lfloor 2k/3 \rfloor$ and $Δ(G)\geq k$ contains a copy of every tree $T$ with $k$ edges and $Δ(T)\leq Δ$. We also prove similar results where alternative degree conditions are considered. For the same class of trees, this verifies exactly a related conjecture of Besomi, Pavez-Signé and Stein, and provides asymptotic confirmations of two others.

Embedding trees using minimum and maximum degree conditions

TL;DR

The paper proves that for large k and trees T with bounded maximum degree Δ, any host graph G with δ(G)≥⌊2k/3⌋ and Δ(G)≥k contains every such T, extending to multiple degree configurations and asymptotically confirming several Besomi–Pavez-Signé–Stein conjectures. The authors develop a sparse-to-dense strategy via a hyperstability-based decomposition into rich subgraphs, then apply regularity- and density-embedded arguments to finish embeddings in a dense subgraph. They obtain exact 2/3 and α-exact results for bounded-degree trees, along with second-neighbour embedding results, and provide a coherent framework for extending sparse embedding problems to dense settings. The work advances the understanding of mixed-degree conditions in tree embeddings and lays groundwork for further exact and asymptotic results in sparse regimes.

Abstract

A variant of the Erdős-Sós conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least and maximum degree at least contains a copy of every tree with edges. Both degree bounds are best possible. We confirm this conjecture for large trees with bounded maximum degree, by proving that for all and sufficiently large , every graph with and contains a copy of every tree with edges and . We also prove similar results where alternative degree conditions are considered. For the same class of trees, this verifies exactly a related conjecture of Besomi, Pavez-Signé and Stein, and provides asymptotic confirmations of two others.

Paper Structure

This paper contains 20 sections, 36 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our results: extensions to the sparse setting
  3. Organisation of the paper
  4. Preliminaries
  5. Proof outline
  6. Basic results
  7. Tree splitting
  8. Regularity
  9. Tools of Besomi, Pavez-Signé and Stein
  10. Converting to the dense setting
  11. Proof of \ref{['thm:approx_k/2']}
  12. Mixed degree combinations: Proof of \ref{['thm:mainalpha-exact']}
  13. Exact $2/3$: Proof of \ref{['thm:exact2/3']}
  14. A rich subgraph has a little more than $k$ vertices
  15. Neighbours in both sides of a bipartite rich subgraph
  16. ...and 5 more sections

Key Result

Theorem 1.2

For all $\varepsilon>0$ there exists $k_0$ such that for all $n$ and $k>k_0$ with $k \geqslant {\varepsilon} n$ the following holds. Every graph on $n$ vertices with minimum degree at least $(2/3+\varepsilon)k$ and maximum degree at least $(1+{\varepsilon})k$ contains a copy of every $k$-edge tree $

Figures (4)

  • Figure 1: Extremal example for \ref{['conj:2k/3']}
  • Figure 2: Forbidden structures in $G$ as detailed in the proof of \ref{['lem:intersectionproperties']}. Note that $x$, $y_1,\dots,y_{2\Delta}$ and $X$ need not actually be disjoint from the rich subgraphs.
  • Figure 3: The tree $T$ on the right is embedded into the structure in $G$ on the left, so that the corresponding colours and placement align. Note that $B_1$ and $B_2$ need not actually be distinct.
  • Figure 4: Embedding process using the $x$-peripheral-matching $M$.

Theorems & Definitions (95)

  • Conjecture 1.1: Havet, Reed, Stein and Wood HRSW02497401
  • Theorem 1.2: Besomi, Pavez-Signé and Stein BPSdegreeconditions2019
  • Conjecture 1.3: Besomi, Pavez-Signé and Stein BPSmaxmin2020
  • Theorem 1.4: Besomi, Pavez-Signé and Stein BPSmaxmin2020
  • Conjecture 1.5: Besomi, Pavez-Signé and Stein BPSdegreeconditions2019
  • Theorem 1.6: Besomi, Pavez-Signé and Stein BPSdegreeconditions2019
  • Conjecture 1.7: Besomi, Pavez-Signé and Stein BPSdegreeconditions2019
  • Theorem 1.8: Besomi, Pavez-Signé and Stein BPSdegreeconditions2019
  • Theorem 1.9
  • Theorem 1.10
  • ...and 85 more