# Connectedness of Certain Graph Coloring Complexes

## TL;DR

This paper investigates the connectedness properties of certain graph coloring complexes, specifically for bipartite graphs formed by the product of K2 and K_n, providing explicit formulas for their connectedness.

## Contribution

It establishes exact connectedness values for Hom complexes of bipartite graphs with complete graphs, extending understanding of their topological structure.

## Key findings

- Connectedness of Hom complexes is m-n-1 for m ≥ n
- Connectedness is m-3 in other cases
- For bipartite graphs K2 × K_n, connectedness is m - d - 2, where d is the maximum degree

## Abstract

In this article, we consider the bipartite graphs $K_2 \times K_n$. We prove that the connectedness of the complex $\displaystyle \text{Hom}(K_2\times K_{n}, K_m) $ is $m-n-1$ if $m \geq n$ and $m-3$ in the other cases. Therefore, we show that for this class of graphs, $\text{Hom} (G, K_m)$ is exactly $m-d-2$ connected, $m \geq n$, where $d$ is the maximal degree of the graph $G$.