Independence complexes of generalized Mycielskian graphs

TL;DR

This work determines the homotopy type of the independence complex of generalized Mycielskian graphs in terms of the independence complex of the base graph and its Kronecker double cover. By analyzing the $l$-Mycielskian $\mu_l(G)$ and the product $G\times P_2$, the authors obtain explicit decompositions as wedges of spaces formed by iterated suspensions and joins of $I(G)$ and $I(G\times P_2)$, with the precise form governed by $l \bmod 3$. They derive corollaries for classical graphs and extend the framework to iterated and iterated Kronecker constructions, yielding wedge-of-spheres descriptions in many cases. The results unify and extend previous findings on Mycielski graphs and enable concrete computations of independence complexes for a broad class of graphs, with implications for topological combinatorics and chromatic-number constructions.

Abstract

We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs.

Figures (2)

• Figure 1: $\mu_l(G)$
• Figure 2: $H$

Theorems & Definitions (33)

• Lemma 1
• Lemma 2
• Proposition 3
• Lemma 4
• proof
• Theorem 5
• proof
• Corollary 6
• Corollary 7
• proof