Homotopy type of the neighborhood complexes of graphs of maximal degree at most $3$ and $4$-regular circulant graphs

TL;DR

This paper investigates the topological structure of neighborhood complexes of certain graphs, revealing their homotopy types and implications for graph coloring bounds.

Contribution

It classifies the homotopy types of neighborhood complexes for graphs with maximum degree 3 and 4-regular circulant graphs, extending topological graph theory.

Findings

Connected components are homotopy equivalent to points, circles, spheres, tori, or connected sums of tori.

Provides explicit topological descriptions for neighborhood complexes of specific graph classes.

Supports Lovász's bounds on chromatic number via topological connectivity.

Abstract

To estimate the lower bound for the chromatic number of a graph $G$, Lov\'asz associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent either to a point, to a wedge sum of circles, to a wedge sum of $2$-spheres $S^2$, to $S^3$, to a garland of $2$-spheres $S^2$ or to a connected sum of tori.