# A proof of a conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp on subgraphs of minimum degree $k$

## TL;DR

This paper proves a conjecture by Erd"H{o}s et al. that adding a single edge beyond a certain threshold guarantees a smaller subgraph with minimum degree at least k, extending previous combinatorial ideas.

## Contribution

It confirms that one extra edge ensures a large enough subgraph with minimum degree k, advancing understanding of degree conditions in graphs.

## Key findings

- Proved Erd"H{o}s et al.'s conjecture on subgraphs with minimum degree k
- Extended ideas of Mousset, Noever, and kori7c to the proof
- Established that a single additional edge guarantees a substantial subgraph

## Abstract

Erd\H{o}s, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k\geq 2$: Every graph on $n\geq k-1$ vertices with at least $(k-1)(n-k+2)+{k-2\choose 2}$ edges contains a subgraph with minimum degree at least $k$. However, there are examples in which the whole graph is the only such subgraph. Erd\H{o}s et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\varepsilon_k)n$ vertices with minimum degree at least $k$, where $\varepsilon_k>0$ depends only on $k$. We prove this conjecture, using and extending ideas of Mousset, Noever and \v{S}kori\'{c}.