The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs
Abstract
The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs M2n and circular ladder graphs 𝒞2n. The Möbius ladder graphs M2n are highly symmetric cubic graphs obtained from a 2n-cycle by joining opposite vertices, while the circular ladder graphs 𝒞2n are the Cartesian products of an n-cycle and a path of length one. We show that Ind(M2n) and Ind(𝒞2n) have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to n modulo $4$. For the perfect matching complex ℳp(M2n), its homotopy type is a wedge of two copies of 𝕊(n−2)/2 when n is even, while for odd n it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on n modulo $6$. The perfect matching complex ℳp(𝒞2n) is contractible for odd n, whereas for even n its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by n modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.
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BibTeX
@misc{agarwal2026homotopy,
title = {{The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs}},
author = {Anshu Agarwal and Biplab Basak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30601},
eprint = {2608.30601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30601},
}
RIS
TY - UNPB
TI - The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs
AU - Anshu Agarwal
AU - Biplab Basak
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.30601
UR - https://arxiv.org/abs/2608.30601
AB - The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs M_2n and circular ladder graphs 𝒞_2n. The Möbius ladder graphs M_2n are highly symmetric cubic graphs obtained from a 2n-cycle by joining opposite vertices, while the circular ladder graphs 𝒞_2n are the Cartesian products of an n-cycle and a path of length one. We show that Ind(M_2n) and Ind(𝒞_2n) have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to n modulo $4$. For the perfect matching complex ℳ_p(M_2n), its homotopy type is a wedge of two copies of 𝕊^(n−2)/2 when n is even, while for odd n it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on n modulo $6$. The perfect matching complex ℳ_p(𝒞_2n) is contractible for odd n, whereas for even n its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by n modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.
ER -