# Two monads on the category of graphs

## TL;DR

This paper introduces two monads on the category of graphs, establishing their Eilenberg-Moore categories as isomorphic to perfect matchings and partial Steiner triple systems, and explores their categorical products.

## Contribution

The paper presents novel monads on graphs and characterizes their Eilenberg-Moore categories as well-known combinatorial structures.

## Key findings

- Eilenberg-Moore categories are isomorphic to perfect matchings and partial Steiner triple systems.
- Provides a categorical framework for understanding these combinatorial structures.
- Describes the product operations in these categories.

## Abstract

We introduce two monads on the category of graphs and prove that their Eilenberg-Moore categories are isomorphic to the category of perfect matchings and the category of partial Steiner triple systems, respectively. As a simple application of these results, we describe the product in the categories of perfect matchings and partial Steiner triple systems.