# (Lack of) Model Structures on the Category of Graphs

## TL;DR

This paper investigates the possibility of defining model structures on the category of finite graphs with -homotopy equivalences, concluding that such structures do not exist under common conditions, highlighting limitations in this categorical framework.

## Contribution

It proves the non-existence of a Strm-Hurewicz type model structure on finite graphs with -homotopy equivalences, revealing fundamental constraints in graph homotopy theory.

## Key findings

- No model structure similar to Strm-Hurewicz exists for finite graphs with -homotopy.
- The category of graphs with -homotopy equivalences lacks a model structure when cofibrations are subclasses of graph inclusions.
- This limitation impacts the development of homotopical methods in graph theory.

## Abstract

In this article, we study model structures on the category of finite graphs with $\times$-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Str\o{}m-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with $\times$-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.