Edge ideals of complements of a class of circular interval graphs

Let $C_n^m$ be the $m$-th power of the cycle on $n$ vertices, and let $\Gnm$ be its complement, where $m\ge1$ and $n\ge3m+1$. We prove that every induced subcomplex of the clique complex of $C_n^m$ is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we est