Homotopy Type of the Neighborhood Complexes of Graphs of Maximal Degree at most $3$ and $4$-Regular Circulant Graphs

Abstract

To estimate the lower bound for the chromatic number of a graph $G$,  Lovász associated a simplicial complex  $\mathcal{N}(G)$ called the neighborhood complex and related 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.