Large induced subgraphs with $k$ vertices of almost maximum degree

Abstract

In this note we prove that for every integer $k$, there exist constants $g_{1}(k)$ and $g_{2}(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at least $n - g_{1}(k)\sqrt{\Delta}$ vertices, such that $H$ has $k$ vertices of the same degree of order at least $\Delta(H)-g_{2}(k)$. This solves a conjecture of Caro and Yuster up to the constant $g_{2}(k)$.