Discussiones Mathematicae Graph Theory 33(3) (2013) 571-582
doi: 10.7151/dmgt.1684

[BIBTex] [PDF] [PS]

The Crossing Numbers of Products of Path with Graphs of Order Six

Marián Klešč and Jana Petrillová

Faculty of Electrical Engineering and Informatics
Technical University of Košice
Letná 9, 042 00 Košice, Slovak Republic

Abstract

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G □ Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G □ Pn for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian product H □ Pn is 2(n −1). In addition, the crossing numbers of G □ Pn for fourty graphs G on six vertices are collected.

Keywords: graph, drawing, crossing number, Cartesian product, path, tree

2010 Mathematics Subject Classification: 05C10, 05C38.

References

[1]L.W. Beineke and R.D. Ringeisen, On the crossing numbers of products of cycles and graphs of order four, J. Graph Theory 4 (1980) 145--155, doi: 10.1002/jgt.3190040203.
[2]D. Bokal, On the crossing number of Cartesian products with paths, J. Combin. Theory (B) 97 (2007) 381--384, doi: 10.1016/j.jctb.2006.06.003.
[3]S. Jendrol' and M. Ščerbová, On the crossing numbers of Sm × Pn and Sm × Cn, Časopis Pro Pěstování Matematiky 107 (1982) 225--230.
[4]M. Klešč, The crossing numbers of Cartesian products of stars and paths or cycles, Math. Slovaca 41 (1991) 113--120.
[5]M. Klešč, The crossing numbers of products of paths and stars with 4-vertex graphs, J. Graph Theory 18 (1994) 605--614.
[6]M. Klešč, The crossing number of K2,3 × Pn and K2,3 × Sn, Tatra Mt. Math. Publ. 9 (1996) 51--56.
[7]M. Klešč, The crossing numbers of products of 4-vertex graphs with paths and cycles, Discuss. Math. Graph Theory 19 (1999) 59--69, doi: 10.7151/dmgt.1085.
[8]M. Klešč, The crossing numbers of Cartesian products of paths with 5-vertex graphs, Discrete Math. 233 (2001) 353--359, doi: 10.1016/S0012-365X(00)00251-X.
[9]D. Kravecová, The crossing number of P25 × Pn, Creat. Math. Inform. 28 (2012) 49--56.
[10]Y.H. Peng and Y.C. Yiew, The crossing number of P(3,1) × Pn, Discrete Math. 306 (2006) 1941--1946, doi: 10.1016/j.disc.2006.03.058.
[11]J. Wang and Y. Huang, The crossing number of K2,4 × Pn, Acta Math. Sci.,Ser. A, Chin. Ed. 28 (2008) 251--255.
[12]L. Zhao, W. He, Y. Liu and X. Ren, The crossing number of two Cartesian products, Int. J. Math. Comb. 1 (2007) 120--127.
[13]W. Zheng, X. Lin, Y. Yang and Ch. Cui, On the crossing number of Km Pn, Graphs Combin. 23 (2007) 327--336, doi: 10.1007/s00373-007-0726-z.

Received 30 November 2011
Revised 12 April 2012
Accepted 17 April 2012