Journal of Applied Mathematics

Volume 2017 (2017), Article ID 7640347, 7 pages

https://doi.org/10.1155/2017/7640347

## 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.

#### 1. Introduction

Let be a simple connected graph with a vertex-set and an edge-set . The crossing number of graph , denoted , is the minimum number of pairwise intersections of edge crossing on the plane drawing of graph . Clearly, if and only if is planar. It is known that the exact crossing numbers of any graphs are very difficult to compute. In 1973, Erdös and Guy [1] wrote,* “Almost all questions that one can ask about crossing numbers remain unsolved.”* In fact, Garey and Johnson [2] prove that computing the crossing number is NP-complete.

A 2-page drawing of is a representation of on the plane such that its vertices are placed on a straight horizontal line according to fixed vertex ordering and its edges are drawn as a semicircle above or below but never cross .

The -*cube* or -*dimensional hypercube * is recursively defined in terms of the Cartesian products. The one-dimension cube is simply where is a complete graph with vertices. For , is defined recursively as . The order of is and its size is . Since is planar for , so for each such . Eggleton and Guy [3] showed that but is unknown for .

It was declared by Eggleton and Guy [3] that the crossing numbers of the hypercube (non-2-page) for was

Then, in 1973, Erdös and Guy [1] conjectured equality in (1). In 1993, a lower bound of was proved by Sýkora and Vrt’o [4]:

In 2008, Faria et al. [5] constructed a new drawing of in the plane which led to the conjectured number of crossings To the best of our knowledge, the fixed linear crossing number for has not been established. In this paper, we discuss a method to obtain an approximation for fixed linear crossing number for hypercube graph.

#### 2. 2-Page Drawings of Hypercube Graph

Throughout this paper, we consider the ordering of hypercube graph . Since is defined recursively as , for , where is a simple graph with 2 vertices together with a single edge incident to both vertices, has 2 copies of with edges connecting between them. Given a fixed ordering on , the vertices of the first are labeled and the vertices of the second are labeled . The two vertices between the first and the second are adjacent if and only if the sum of the labeled is . Figures 1 and 2 present the ordering of and which we consider throughout this paper. Notice that our method is independent on vertex ordering; therefore, for a fixed , we can apply the method times so as to obtain the 2-page linear crossing number.