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Journal of Applied Mathematics
Volume 2017 (2017), Article ID 7640347, 7 pages
https://doi.org/10.1155/2017/7640347
Research Article

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

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to T. Mouktonglang

Received 5 January 2017; Revised 20 March 2017; Accepted 10 April 2017; Published 8 May 2017

Academic Editor: Heping Zhang

Copyright © 2017 A. Suebsriwichai and T. Mouktonglang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.