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In this paper and a companion paper, we prove that, if $m$ is sufficientlylarge, every graph on $m+1$ vertices that has a universal vertex and minimumdegree 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 specialcase of a recent conjecture by Havet, Reed, Stein, and Wood. The present paperalready contains an approximate version of the result.

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