Homotopy Type of Total Cut Complexes of Squared Cycle Graphs

TL;DR

The paper advances the understanding of total cut complexes for squared cycle graphs by proving the conjectured homotopy types in key cases: a complete description for k=3 and partial results for general k when n equals 3k+1 or 3k+2. It develops a detailed discrete Morse-theoretic framework to describe faces and construct acyclic matchings, yielding explicit sphere decompositions such as Δ_3^t(W_n) ≃ S^{2n-17} for n=9,10 and ≃ S^{n-6} for n≥11, and proving Δ_k^t(W_{3k+1}) ≃ S^3, Δ_k^t(W_{3k+2}) ≃ S^5 in the general case. The work connects total cut complexes to neighborhood complexes of stable Kneser graphs and lays groundwork for generalized conjectures, supported by SageMath data on total cut complexes of powers of cycles. These results extend the Fröberg-type landscape in topological combinatorics and suggest broad applicability of the Morse-theoretic approach to more graph families.

Abstract

In this paper, we investigate the homotopy type and combinatorial properties of total cut complexes of squared cycle graphs. The total cut complexes are a new type of graphical complexes introduced by Bayer et al.(2024) to extend Fröberg's theorem. In Bayer et al.[Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2): 1630--1675 (2024)], the authors made a conjecture on the homotopy type of total cut complexes of squared cycle graphs for $k \geq 3$. We proved this conjecture in the case when $k=3$ . For general $k\geq 3$, we confirmed the cases when $n =3k+1$ and $3k+2$.

Theorems & Definitions (23)

• Theorem 1.1: Froberg_1990, Eagon_1998
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4: Bayer_2024_02,Conjecture 5.1
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4