Spanning trees in graphs of high minimum degree with a universal vertex I: An asymptotic result

In this paper and a companion paper, we prove that, ifm $m$ is sufficiently large, every graph onm + 1 $m+1$ vertices that has a universal vertex and minimum degree at least⌊2 m 3 ⌋$\lfloor \phantom{\rule[-0.5em]{}{0ex}}\frac{2m}{3}\rfloor $ contains each treeT $T$ withm $m$ edges as a subgraph. Our result confirms, for largem $m$ , 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.