# Spanning Trees in Graphs of High Minimum Degree with a Universal Vertex I: An Asymptotic Result

## TL;DR

This paper proves that large graphs with a universal vertex and high minimum degree contain all trees with a given number of edges, confirming a special case of a recent conjecture.

## Contribution

It establishes an asymptotic result for spanning trees in graphs with a universal vertex and high minimum degree, advancing understanding of tree embeddings.

## Key findings

- Graphs with a universal vertex and high minimum degree contain all spanning trees with a given number of edges.
- The result confirms a special case of a conjecture by Havet, Reed, Stein, and Wood for large graphs.
- An approximate version of the main theorem is provided in the paper.

## Abstract

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result.