Independence complexes of generalized Mycielskian graphs

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.

💡 Research Summary

The paper investigates the topological structure of independence complexes of graphs obtained by two well‑known constructions: the generalized Mycielskian µₗ(G) and the Kronecker double cover G ⊠ P₂. The independence complex I(G) of a simple graph G is the simplicial complex whose simplices are the independent vertex sets of G. The authors first recall two elementary lemmas: (1) the independence complex of a disjoint union G₁ ∪ G₂ splits as a join I(G₁) ∗ I(G₂); (2) if the closed neighborhoods of two distinct vertices satisfy N(v) ⊆ N(u), then deleting u does not change the homotopy type of I(G). A key technical tool (Proposition 3) shows that when the inclusion I(G − N(v)) → I(G − v) is null‑homotopic, the complex I(G) is homotopy equivalent to I(G − v) ∨ Σ I(G − N(v)).

The generalized Mycielskian µₗ(G) is defined by taking the categorical product G ⊠ P_{ℓ+1}, adding a new apex vertex w adjacent to all vertices of the first copy of G, and then connecting each level i (1 ≤ i ≤ ℓ) to level i+1 according to the edges of G. This construction yields a hierarchy of vertex layers V₁, V₃, V₆, …, V_{3k} whose neighborhoods satisfy N(v, i) ⊆ N(v, i+3). By repeatedly applying Lemma 2 to delete the higher layers, the authors reduce I(µₗ(G)) to a combination of the original complex I(G) and the complex of the Kronecker double cover I(G ⊠ P₂).

The main result, Theorem 5, distinguishes three residue classes of ℓ modulo 3. For ℓ = 3k the complex splits as a wedge of I(G) ∗ I(G ⊠ P₂)^{∗k} and Σ I(G ⊠ P₂)^{∗k}. For ℓ = 3k+1 an extra copy of I(G ⊠ P₂)^{∗k} appears, and for ℓ = 3k+2 an additional copy of I(G ⊠ P₂)^{∗k} is added after a suspension. The proof proceeds by an inductive deletion of the vertex layers, using Lemma 2 to guarantee homotopy equivalences and Proposition 3 to introduce suspensions when necessary.

The paper then turns to the Kronecker double cover G ⊠ P₂. Theorem 14 gives the homotopy type of I(µₗ(G) ⊠ P₂) in terms of suspensions of I(G ⊠ P₂). The authors introduce two auxiliary functions f(k,r) and g(k,r) (Lemma 4) that count the number of suspensions appearing after r iterations of the Mycielskian. Using these functions they obtain explicit formulas for the independence complexes of iterated Mycielskians µᵣ(µₗ(G)) (Corollary 15, Theorem 16, Theorem 17). In particular, for ℓ = 3k+1 and ℓ = 3k+2 the iterated complexes are wedges of many copies of I(G) and I(G ⊠ K₂), each possibly suspended several times.

The authors apply the general theory to several families of graphs. For paths Pₙ they compute the exact wedge of spheres (Corollary 11), distinguishing the three residue classes of ℓ. For cycles Cₙ they provide a table (Table 1) listing the dimensions and multiplicities of spheres appearing in I(µₗ(Cₙ)). For complete graphs Kₙ and their categorical product Kₙ ⊠ Kₘ they recover known results (Corollary 7, 8) and extend them to the generalized Mycielskian. When G is bipartite, Proposition 10 shows that I(µₗ(G)) is a wedge of suspensions of I(G) and Σ I(G) (the latter coming from the double cover). Consequently, for forests, rectangular lattices with at most six rows, and other bipartite graphs the independence complex of the generalized Mycielskian is either contractible or a wedge of spheres (Corollaries 12‑13, 18‑20).

Overall, the paper establishes a robust framework: the homotopy type of the independence complex of µₗ(G) is completely determined by the homotopy types of I(G) and I(G ⊠ P₂). This reduces the study of complex graph constructions to the analysis of two relatively simple complexes. The results have immediate implications for graph coloring (since Mycielski constructions raise chromatic number without creating triangles), for the topology of graph configuration spaces, and for combinatorial topology more broadly. The explicit formulas for paths, cycles, and products of complete graphs provide concrete examples that illustrate the power of the method and open the way for further investigations into more intricate graph families.