Embedding trees using minimum and maximum degree conditions

TL;DR

This paper proves a conjecture about embedding large trees with bounded maximum degree into graphs with specific minimum and maximum degree conditions, confirming related conjectures and extending previous results.

Contribution

It confirms the Havet-Reed-Steiner conjecture for large bounded-degree trees and degree conditions, extending the understanding of tree embeddings in graphs.

Findings

Graphs with specified degree bounds contain all large bounded-degree trees.

The results verify conjectures of Besomi, Pavez-Signé, and Stein.

Provides asymptotic confirmations of related conjectures.

Abstract

A variant of the Erd\H{o}s-S\'os 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 $\Delta\in \mathbb{N}$ and sufficiently large $k\in \mathbb{N}$, every graph $G$ with $\delta(G)\geq \lfloor 2k/3 \rfloor$ and $\Delta(G)\geq k$ contains a copy of every tree $T$ with $k$ edges and $\Delta(T)\leq \Delta$. 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-Sign\'e and Stein, and provides asymptotic confirmations of two others.