On the crossing number of K ( 9 , 9 ) SURF 2002 Final Report

Because of the large success of very large scale integration (VLSI) technology many researchers have focused on optimizing the VLSI circuit layout. One of the major tasks is minimizing the number of wire crossings in a circuit, as this greatly reduces the chance of cross-talk in long crossing wires carrying the same signal and also allows for faster operation and less power dissipation. The question of finding the minimal number of crossing wires can be abstracted to a graph theoretical problem of determining the minimal number of edge crossings in a drawing of a given graph. The crossing number problem is especially interesting for complete bipartite graphs, for which Zarankiewicz conjectured a formula in 1954 that still remains unproven. In 1993 Woodall used a computer program to solve the smallest then unknown case that of K(7, 7) thus proving the Zarankiewicz conjecture for K(m, n) with min(m, n) ≤ 8. The smallest now unsolved case is that of K(9, 9). The purpose of this project is to write a program that reproduces Woodall’s results and further checks the conjecture for K(9, 9).