Let (Cmd)∞ denote the graph whose set of vertices is Zmd in which two distinct vertices are adjacent iff in each coordinate either they are equal or they differ, modulo m, by at most 1. Bollobás, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a set of vertices of (Cm d)∞ whose deletion destroys all topologically nontrivial cycles is md - (m - 1)d. We present a short proof of this result, using the Brunn-Minkowski inequality, and also show that the bound can be achieved only by selecting a value xi in each coordinate i, 1 ≤ i ≤ d, and by keeping only the vertices whose ith coordinate is not xi for all i.
All Science Journal Classification (ASJC) codes
- Brunn-Minkowski inequality
- Discrete torus
- Nontrivial cycles