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

## TL;DR

This paper proves that large induced subgraphs with many vertices of nearly maximum degree exist in graphs, confirming a conjecture up to a constant factor, and provides bounds based on maximum degree and vertex count.

## Contribution

It establishes the existence of large induced subgraphs with specified degree properties, solving a conjecture by Caro and Yuster up to a constant.

## Key findings

- Existence of induced subgraphs with many vertices of almost maximum degree
- Bounds depend on maximum degree and number of vertices
- Confirms a conjecture of Caro and Yuster up to a constant

## 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 $g_{2}(k)$.