On the independence number of the strong product of cycle-powers

We determine the independence number of the strong product of cycle-powers C-n(k) and C-m(p), where C-n(k) denotes the graph obtained from the n-cycle C-n by adding all chords joining vertices at most k steps apart on the cycle. The result generalizes a similar result for odd cycles obtained by Hales. The solution is based on the problem of arranging t 1s and m - t Os in a circle (where t = left perpendicularmk/pright perpendicular) in such a way that every string of p consecutive bits has at most k equal to 1. A nontrivial lower bound for the Shannon capacity of cycle-powers is obtained on the basis of the independence numbers computed. The result can also be interpreted in terms of packing rectangles into a torus. The maximum number of p-by-k rectangles that can be packed into a two-dimensional m-by-n (rectangular) torus is obtained. The proof of the main theorem can be used to determine the maximum packing itself (or the corresponding largest independent set in the product graph). (C) 2012 Elsevier B.V. All rights reserved.