Embedding of Hypercube into Cylinder

Task mapping in modern high performance parallel computers can be modeled as a graph embedding problem, which simulates the mapping as embedding one graph into another and try to find the minimum wirelength for the mapping. Though embedding problems have been considered for several regular graphs, such as hypercubes into grids, binary trees into grids, et al, it is still an open problem for hypercubes into cylinders. In this paper, we consider the problem of embedding hypercubes into cylinders to minimize the wirelength. We obtain the exact wirelength formula of embedding hypercube $Q^r$ into cylinder $C_{2^3}\times P_{2^{r-3}}$ with $r\ge3$.