Clique problem, cutting plane proofs and communication complexity

Motivated by its relation to the length of cutting plane proofs for the Maximum Biclique problem, we consider the following communication game on a given graph G, known to both players. Let K be the maximal number of vertices in a complete bipartite subgraph of G, which is not necessarily an induced subgraph if G is not bipartite. Alice gets a set A of vertices, and Bob gets a disjoint set B of vertices such that |A|+|B|>K. The goal is to find a nonedge of G between A and B. We show that O(\log n) bits of communication are enough for every n-vertex graph.
10 pages. Theorem 1 in the previous version holds only for bipartite graphs, the non-bipartite case remains open. I now separate the bipartite and non-bipartite cases (by switching from independent sets to cliques, hence a new title). Some new open problems as well as references are added