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

Abstract EN

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.