SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2108-2123, January 2020.
We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ 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 $\frac k2$ and $\frac 43k$, respectively, as long as the host graph avoids a specific structure.

中文翻译：

---

嵌入树的最大和最小度条件

SIAM离散数学杂志，第34卷，第4期，第2108-2123页，2020年1月。
我们提出以下猜想：对于每个固定的\\ alpha \ in [0，\ frac 13）$，每个最小度数图至少$（1+ \ alpha）\ frac k2 $和最大度至少为$ 2（1- \ alpha）k $包含每棵具有$ k $边的树作为子图。我们的主要结果是有界度树和大型密集主体图的猜想的近似版本。我们还表明，我们的猜想在渐近上是最好的。近似结果的证明依赖于第二个结果，我们认为它本身很有趣。即，只要宿主图避免特定的结构，我们就可以将任何有界度树分别嵌入到渐近超过$ \ frac k2 $和$ \ frac 43k $的最小/最大程度的宿主图中。