Maximum and minimum degree conditions for embedding trees

Abstract

We propose the following conjecture: For every fixed alpha is an element of [0, 1/3), each graph of minimum degree at least (1 + alpha)k/2 and maximum degree at least 2(1 - alpha)k contains each tree with k edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asymptotically exceeding k/2 and 4/3 k, respectively, as long as the host graph avoids a specific structure.