An upper bound for the crossing number of augmented cubes

A good drawing of a graph G is a drawing where the edges are non-self-intersecting and each of the two edges have at most one point in common, which is either a common end vertex or a crossing. The crossing number of a graph G is the minimum number of pairwise intersections of edges in a good drawing of G in the plane. The n-dimensional augmented cube AQ n , proposed by S.A. Choudum and V. Sunitha [Augmented cubes, Networks 40 (2002), pp. 71–84], is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of AQ n less than .

[1]  Jun-Ming Xu,et al.  Fault-tolerant pancyclicity of augmented cubes , 2007, Inf. Process. Lett..

[2]  L. Faria,et al.  An improved upper bound on the crossing number of the hypercube , 2008 .

[3]  Lowell W. Beineke,et al.  The crossing number of C3 × Cn , 1978, J. Comb. Theory, Ser. B.

[4]  Tom Madej,et al.  Bounds for the crossing number of the N-cube , 1991, J. Graph Theory.

[5]  Sun-Yuan Hsieh,et al.  Conditional Diagnosability of Augmented Cubes under the PMC Model , 2012, IEEE Transactions on Dependable and Secure Computing.

[6]  Ondrej Sýkora,et al.  On crossing numbers of hypercubes and cube connected cycles , 1993, BIT.

[7]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[8]  Sun-Yuan Hsieh,et al.  Conditional edge-fault Hamiltonicity of augmented cubes , 2010, Inf. Sci..

[9]  Frank Thomson Leighton,et al.  A Framework for Solving VLSI Graph Layout Problems , 1983, J. Comput. Syst. Sci..

[10]  Jun-Ming Xu,et al.  The forwarding indices of augmented cubes , 2007, Inf. Process. Lett..

[11]  Dharma P. Agrawal,et al.  Generalized Hypercube and Hyperbus Structures for a Computer Network , 1984, IEEE Transactions on Computers.

[12]  Jimmy J. M. Tan,et al.  Fault hamiltonicity of augmented cubes , 2005, Parallel Comput..

[13]  Sheshayya A. Choudum,et al.  Augmented cubes , 2002, Networks.

[14]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[15]  Sun-Yuan Hsieh,et al.  Strong Diagnosability and Conditional Diagnosability of Augmented Cubes Under the Comparison Diagnosis Model , 2012, IEEE Transactions on Reliability.

[16]  Shi-Jinn Horng,et al.  Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes , 2009, Appl. Math. Comput..

[17]  Jimmy J. M. Tan,et al.  Embedding Hamiltonian paths in augmented cubes with a required vertex in a fixed position , 2009, Comput. Math. Appl..

[18]  David S. Johnson,et al.  Crossing Number is NP-Complete , 1983 .

[19]  Pao-Lien Lai,et al.  Geodesic pancyclicity and balanced pancyclicity of Augmented cubes , 2007, Inf. Process. Lett..

[20]  Shengjun Pan,et al.  The crossing number of K11 is 100 , 2007, J. Graph Theory.

[21]  Yuansheng Yang,et al.  The crossing numbers of generalized Petersen graphs with small order , 2009, Discret. Appl. Math..

[22]  P. Erdös,et al.  Crossing Number Problems , 1973 .

[23]  Sun-Yuan Hsieh,et al.  Cycle embedding of augmented cubes , 2007, Appl. Math. Comput..

[24]  F. Thomas Leighton,et al.  Complexity Issues in VLSI , 1983 .

[25]  R. Bruce Richter,et al.  The crossing number of c4 × c4 , 1995, J. Graph Theory.

[26]  W. T. Tutte.  Toward a theory of crossing numbers , 1970 .

[27]  Celina M. H. de Figueiredo,et al.  An Improved Upper Bound on the Crossing Number of the Hypercube , 2003, WG.

[29]  Paul Turán,et al.  A note of welcome , 1977, J. Graph Theory.

[30]  Frank Thomson Leighton.  Introduction to parallel algorithms and architectures: arrays , 1992 .

[31]  Frank Thomson Leighton,et al.  New lower bound techniques for VLSI , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[32]  Martin Skoviera,et al.  ON EGGLETON AND GUY'S CONJECTURED UPPER BOUND FOR THE CROSSING NUMBER OF THE n-CUBE , 2000 .

[33]  Gelasio Salazar.  On the crossing numbers of loop networks and generalized Petersen graphs , 2005, Discret. Math..

[34]  Carsten Thomassen,et al.  Intersections of curve systems and the crossing number ofC5×C5 , 1995, Discret. Comput. Geom..

[35]  Jun-Ming Xu,et al.  Panconnectivity and edge-fault-tolerant pancyclicity of augmented cubes , 2007, Parallel Comput..