# Smaller subgraphs of minimum degree k

## TL;DR

This paper advances the understanding of subgraphs with minimum degree k by showing that a linear number of vertices can be removed while maintaining the degree condition, moving closer to a longstanding conjecture.

## Contribution

The authors improve existing bounds by demonstrating that at least a logarithmic fraction of vertices can be removed to preserve minimum degree k, supporting Erdős et al.'s conjecture.

## Key findings

- At least rac{n}{\u221a{ ext{log} n}} vertices can be removed.
- Progress towards Erdf3s et al.'s conjecture on subgraph size.
- Enhanced bounds on subgraph vertex removal for minimum degree k.

## Abstract

In 1990 Erd\H{o}s, Faudree, Rousseau and Schelp proved that for $k\geq 2$, every graph with $n\geq k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n}/\sqrt{6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $\epsilon_k>0$. We make progress towards their conjecture by showing that one can remove at least $\Omega(n/\log n)$ many vertices.