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

Abstract

We determine the independence number of the strong product of cycle-powers Cnk and Cmp, where Cnk denotes the graph obtained from the n-cycle Cn 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 0s in a circle (where t=⌊mk/p⌋) 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).