On independence complexes of graph products

TL;DR

The paper tackles the problem of determining the homotopy types of independence complexes $I(G)$ for graph products where one factor is a path, and extends the analysis to induced subgraphs and lexicographic/strong products. It develops decomposition techniques for $I(G\times P_n)$ in terms of $I(G\times P_2)$, and analyzes lexicographic and strong products through polyhedral joins, star-cluster arguments, and vertex-deletion lemmas, yielding explicit homotopy types in many cases. It establishes that $I(P_n\times P_m)$ and related subgraphs are contractible or wedges of spheres, and proves $P_n\times P_m\in\mathcal{SW}$ for $m\le 4$ (with analogous results for strong products), accompanied by recurrences and generating functions for Betti numbers. The work provides a structured framework for understanding independence complexes under key graph-product operations and highlights open questions for general parameter ranges, suggesting natural extensions via suspensions and further product families.

Abstract

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$ and $P_n\boxtimes P_4$. We also focus in the independence complexes of the induced subgraphs of $P_n\times P_3$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$, $P_n\boxtimes P_4$ and some lexicographic products $G\circ H$.

Figures (4)

• Figure 1: $Q_4$
• Figure 2: Auxiliary graphs used in Theorem \ref{['theosubgrapdstrongprod']} (I).
• Figure 3: Auxiliary graphs used in Theorem \ref{['theosubgrapdstrongprod']} (II).
• Figure 4: Auxiliary graphs used in Theorem \ref{['theosubgrapdstrongprod']} (III).

Theorems & Definitions (46)

• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• Proposition 7
• proof
• Lemma 8