Strong Isometric Dimension, Biclique Coverings, and Sperner's Theorem

Abstract

The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's theorem, the strong isometric dimension of the Hamming graphs K 2 K n is determined.

A set coverage problem

“…De Caen, Gregory and Pullman [2] proved that br(Ī n ) = s(n), whereĪ n is the complement of the identity matrix I n , and s(n) is the smallest integer k such that n k k 2 . The same result was recently rediscovered in [1,5] (see previous paragraph), with a new proof, for biclique covering of the bipartite graph K − n,n , where K − n,n is the complete bipartite graph K n,n with a perfect matching removed.…”

Nondeterministic Communication Complexity of Random Boolean Functions (Extended Abstract)

“…where the first two "o(1)"s follow from (14) and Lemma 5, and the third is the same calculation as in the previous case.…”

The Biclique Covering Number of Grids

“…Two more classes of graphs for which we know the biclique covering number exactly are K − 2n and K − n,n , which are the graphs obtained from K 2n and K n,n by deleting the edges of a perfect matching. The biclique covering number of K − 2n is log 2 n (see [11]) and the biclique covering number of K − n,n is the smallest k for which n k k /2 (see [6,3]). In this paper, we determine the biclique covering number for all grids.…”