# Thresholds in random motif graphs

## TL;DR

This paper generalizes the Erdős-Rényi model to include fixed motifs, establishing thresholds for key properties like connectivity and Hamiltonicity, and providing hitting time results in the random motif graph process.

## Contribution

It introduces the binomial random motif graph model and determines thresholds for several fundamental graph properties, extending classical results to motif-based random graphs.

## Key findings

- Thresholds for connectivity, Hamiltonicity, and perfect matching are established.
- Hitting time results for the emergence of these properties are proved.
- The model generalizes Erdős-Rényi graphs by incorporating fixed motifs.

## Abstract

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding an instance of a fixed graph $H$ on each of the copies of $H$ in the complete graph on $n$ vertices, independently with probability $p$. We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance. Moreover, in the first three cases we give the analogous hitting time results; with high probability, the first graph in the random motif graph process that has minimum degree one (or two) is connected and contains a perfect matching (or Hamiltonian respectively).