# A note on spanning $K_r$-cycles in random graphs

## TL;DR

This paper determines the threshold probability for the appearance of edge-disjoint $K_r$-cycles covering all vertices in a random graph, providing a concise proof leveraging recent results.

## Contribution

It establishes the exact threshold for spanning $K_r$-cycles in random graphs using a simplified proof based on Riordan's recent work.

## Key findings

- Identifies the threshold probability for spanning $K_r$-cycles in $G_{n,p}$.
- Provides a concise proof of the main result.
- Enhances understanding of cycle structures in random graphs.

## Abstract

We find the threshold for the existence of a collection of edge disjoint copies of $K_r$ that form a cyclic structure and span all vertices of $G_{n,p}$. We use a recent result of Riordan to give a two line proof of the main result.