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 establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of $R/I(\Gnm)$. We also prove that $R/I(\Gnm)$ is a Buchsbaum ring of depth $2$ and that the independence complex of $\Gnm$ is a triangulated $m$-manifold with boundary for $m\ge2$, namely a Möbius band or an annulus when $m=2$. For the complement $\overline H$ of every graph $H$ in the above class, we show that $\reg(I(\overline H)^k)=2k+\indm(\overline H)-1$ for all $k\ge2$, where $\indm$ denotes the induced matching number. In particular, some power of $I(\Gnm)$ has a linear resolution if and only if $n\ge4m+1$. In this range, we further prove that the colon ideals $(I^{k+1}:M)$, where $I=I(\Gnm)$ and $M$ is a minimal monomial generator of $I^k$, have regularity $2$.