On the neighborhood complex of the circular complete graphs (Q6606290)

In this article, the author rigorously investigates the homotopy types of specific subposets \(C_{m,n}\) in the Boolean lattice, defined by ``tight'' \(n\)-subsets of \([m]\). These subposets are intricately connected to the neighborhood complexes of circular complete graphs \(K_p^q\). The paper provides a detailed proof establishing the equivalence between \(C_{m,n}\) and the neighborhood complex of \(K_p^q\) for carefully chosen parameters, thus bridging topological methods with graph-theoretical problems.\N\NA significant portion of the paper is dedicated to analyzing the chromatic number of \(K_p^q\) via the Lovász bound. By leveraging the homotopy equivalence of these subposets to neighborhood complexes, Osztényi evaluates the sharpness of Lovász's lower bound. The author applies advanced combinatorial topology tools to characterize the homotopy types of the complexes, which serve as critical invariants in understanding graph colorings and topological obstructions.