Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation

Abstract

The crossing number of graph G is the minimum number of edges crossing in any drawing of G in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of G. We consider a conflict graph G′ of G. Then, instead of minimizing the crossing number of G, we show that it is equivalent to maximize the weight of a cut of G′. We formulate the original problem into the MAXCUT problem. We consider a semidefinite relaxation of the MAXCUT problem. An example of a case where G is hypercube is explicitly shown to obtain an upper bound. The numerical results confirm the effectiveness of the approximation.