# An Erd\H{o}s-Gallai-type theorem for keyrings

## TL;DR

This paper proves a new Erdős-Gallai-type theorem establishing that graphs with sufficiently high average degree contain specific keyring subgraphs with leaves and a minimum number of edges.

## Contribution

It introduces a novel theorem characterizing the existence of keyring subgraphs in graphs based on average degree constraints.

## Key findings

- Graphs with average degree > k-1 contain keyrings with r leaves and at least k edges.
- The theorem applies for all r ≤ (k-1)/2.
- Provides a new extremal condition for keyring subgraphs.

## Abstract

A keyring is a graph obtained by appending $r \geq 1$ leaves to one of the vertices of a cycle. We prove that for every $r \leq (k-1)/2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$ edges.