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It was conjectured by Goyal, Shukla and Singh that the independence complexof the categorical product $K_2\times K_3\times K_n$ has the homotopy type of awedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjectureby calculating the homotopy type of the independence complex of the graphs$C_{3r}\times K_n$ and $K_2\times K_m\times K_n$. For $C_m \times K_n$ when $m$is not a multiple of $3$, we calculate the homotopy type for $m = 4, 5$ andshow that for other values it has to have the homotopy type of a wedge ofspheres of at most $2$ consecutive dimensions and maybe some Moore spaces.

Homotopy type of the independence complex of some categorical products of graphs