The Homotopy Types of the Independence and Perfect Matching Complex of Möbius Ladder Graph

Source abstract

The independence complex and the perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining the homotopy types and other topological properties of these complexes is, in general, a difficult problem, and explicit descriptions are known only for relatively restricted classes of graphs, typically possessing substantial combinatorial regularity or symmetry. Although these complexes have been extensively investigated for several families of graphs, including square grid graphs, comparatively little is known for other natural families of highly symmetric graphs. In this article, we determine the homotopy types of the independence complex and the perfect matching complex of the Möbius ladder graphs $M_{2n}$. Möbius ladder graphs form a natural and highly symmetric family of cubic graphs obtained from a $2n$-cycle by joining pairs of opposite vertices. We prove that the homotopy type of the independence complex $\operatorname{Ind}(M_{2n})$ depends periodically on $n$ modulo $4$: for $n=4k$, it is homotopy equivalent to $\mathbb{S}^{2k-1}$; for $n=4k+2$, it is homotopy equivalent to a wedge of three copies of $\mathbb{S}^{2k}$; and for $n=4k+1$ or $4k+3$, it is homotopy equivalent to $\mathbb{S}^{2k}$. We further determine the homotopy type of the perfect matching complex $\mathcal{M}_p(M_{2n})$. For even $n$, it is homotopy equivalent to a wedge of two copies of $\mathbb{S}^{(n-2)/2}$, while for odd $n$ its homotopy type depends periodically on $n$ modulo $6$. Thus, our results provide explicit descriptions of two fundamental simplicial complexes associated with an important family of highly symmetric cubic graphs.