Homotopy Type of Total Cut Complexes of Squared Cycle Graphs

TL;DR

This paper studies the homotopy type of total cut complexes of squared cycle graphs, confirming a conjecture for specific cases and advancing understanding of their topological properties.

Contribution

It proves the conjecture for squared cycle graphs with k=3 and confirms cases for n=3k+1 and 3k+2, extending the theory of total cut complexes.

Findings

Confirmed the homotopy type conjecture for k=3.

Validated cases for n=3k+1 and 3k+2.

Extended the understanding of total cut complexes' topology.

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 Fr\"oberg'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 $3k+2$.