Spanning trees with large maximum degrees

Abstract

The celebrated result of Komlós, Sárközy, and Szemerédi states that for any $\epsilon >0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $\hat{I}´(G)\ge (1/2+\epsilon )n$ contains every $n$-vertex tree with maximum degree at most $cn/\log n$. This is best possible up to the value of $c$. In this paper, we extend this result to trees with higher maximum degrees, and prove that for $\hat{I}”\gg n/\log n$, roughly speaking, $\hat{I}´(G)\ge n-n^{1-(1+o(1))\hat{I}”/n}$ is the asymptotically optimal minimum degree condition which guarantees that $G$ contains every $n$-vertex spanning tree with maximum degree at most $\hat{I}”$. We also prove the corresponding statements in the random graph setting.

9 pages