Counterexamples in AB percolation
Tomasz Łuczak, John C. Wierman · Journal of Physics A Mathematical and General · 1989
Halley (1983) conjectured that the AB percolation probability function attains its maximum at p=1/2, and that consequently infinite AB percolation is impossible on a bipartite graph whose standard site percolation critical probability is greater than 1/2. The authors produce counterexamples that demonstrate that both conjectures are false. In fact, for any positive integer N, a graph may be constructed which has at least N AB percolation transitions.