Quillen model structures on the category of graphs

TL;DR

This paper explores various Quillen model structures on a specific graph category, demonstrating how graph cores relate to homotopy types and establishing the existence of numerous such structures.

Contribution

It introduces multiple Quillen model structures for a graph category, linking graph cores to homotopy types and showing the diversity of these structures.

Findings

Graph cores correspond to homotopy types in a Quillen model structure

Multiple distinct Quillen model structures exist for the graph category

The paper provides methods to endow graphs with model structures

Abstract

We present different ways of endowing a particular category of graphs with Quillen model structures. We show, among other things, that the core of a graph can be seen as its homotopy type in an appropriate Quillen model structure, and that an infinity of Quillen model structures exist for our particular category of graphs.