### Degree Sequences of $F$-Free Graphs

#### Abstract

Let $F$ be a fixed graph of chromatic number $r+1$. We prove that for all large $n$ the degree sequence of any $F$-free graph of order $n$ is, in a sense, close to being dominated by the degree sequence of some $r$-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better estimates and allows $F$ to grow with $n$.

As an application of our theorem, we present new results on the generalization of the Turán problem introduced by Caro and Yuster [*Electronic J. Combin.* **7** (2000)].