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

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

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