# Haj\'os-Type Constructions and Neighborhood Complexes

## TL;DR

This paper explores Hajós-type graph constructions and their impact on neighborhood complexes, revealing restrictions for highly-connected complexes and introducing algorithms with computational experiments.

## Contribution

It demonstrates the frequent emergence of $S^1$-wedge summands in neighborhood complexes during Hajós-type constructions and introduces new algorithms for graph construction.

## Key findings

- Presence of $S^1$-wedge summands in neighborhood complexes
- Restrictions on construction sequences for highly-connected complexes
- Development and testing of new graph construction algorithms

## Abstract

Any graph $G$ with chromatic number $k$ can be constructed by iteratively performing certain graph operations on a sequence of graphs starting with $K_k$, resulting in a variety of Haj\'os-type constructions for $G$. Finding such constructions for a given graph or family of graphs is a challenging task. We show that the basic steps in these Haj\'os-type constructions frequently result in the presence of an $S^1$-wedge summand in the neighborhood complex of the resulting graph. Our results imply that for a graph $G$ with a highly-connected neighborhood complex, the end behavior of the construction sequence is quite restricted, and we investigate these restrictions in detail. We also introduce two graph construction algorithms based on different Haj\'os-type constructions and conduct computational experiments using these.