Skip to main content Accesibility Help



The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$ , and the star graph, denoted $K_{1,m}$ , is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$ , and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$ . An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

Corresponding author
Hide All
[1] Ábrego, B. M., Aichholzer, O., Fernández-Merchant, S., Hackl, T., Pammer, J., Pilz, A., Ramos, P., Salazar, G. and Vogtenhuber, B., ‘All good drawings of small complete graphs’, in: Proc. 31st European Workshop on Computational Geometry (EuroCG) 2015, Book of Abstracts ,, 57–60.
[2] Anderson, M., Richter, R. B. and Rodney, P., ‘The crossing number of C 6 × C 6 ’, Congr. Numer. 117 (1996), 97107.
[3] Anderson, M., Richter, R. B. and Rodney, P., ‘The crossing number of C 7 × C 7 ’, Congr. Numer. 125 (1996), 97117.
[4] Asano, K., ‘The crossing number of K 1, 3, n and K 2, 3, n ’, J. Graph Theory 10 (1986), 18.
[5] Beineke, L. W. and Ringeisen, R. D., ‘On the crossing numbers of products of cycles and graphs of order four’, J. Graph Theory 4(2) (1980), 145155.
[6] Bokal, D., ‘On the crossing number of Cartesian products with paths’, J. Combin. Theory Ser. B 97(3) (2007), 381384.
[7] Cabello, S. and Mohar, B., ‘Adding one edge to planar graphs makes crossing number and 1-planarity hard’, SIAM J. Comput. 42(5) (2013), 18031829.
[8] Chimani, M. and Wiedera, T., ‘An ILP-based proof system for the crossing number problem’, in: 24th European Symposium on Algorithms (ESA 2016), Aarhus, Denmark, Leibniz International Proceedings in Informatics, 56 (Schloss Dagstuhl, Dagstuhl, Germany, 2016), 29.129.13.
[9] Clancy, K., Haythorpe, M. and Newcombe, A., ‘An effective crossing minimisation heuristic based on star insertion’, preprint available at arXiv:abs/1804.09900.
[10] Dean, A. M. and Richter, R. B., ‘The crossing number of C 4 × C 4 ’, J. Graph Theory 19(1) (1995), 125129.
[11] Garey, M. R. and Johnson, D. S., ‘Crossing number is NP-complete’, SIAM J. Algebr. Discrete Methods 4(3) (1983), 312316.
[12] Glebsky, L. Y. and Salazar, G., ‘The crossing number of C m × C n is as conjectured for nm (m + 1)’, J. Graph Theory 47(1) (2004), 5372.
[13] Harary, F., Kainen, P. C. and Schwenk, A. J., ‘Toroidal graphs with arbitrarily high crossing numbers’, Nanta Math. 6(1) (1973), 5867.
[14] Jendrol, S. and Šcerbová, M., ‘On the crossing numbers of S m × P n and S m × C n ’, Casopis pro Pestováni Mat. 107 (1982), 225230.
[15] Klešč, M., ‘On the crossing numbers of Cartesian products of stars and paths or cycles’, Math. Slovaca 41(2) (1991), 113120.
[16] Klešč, M. and Kravecová, D., ‘The crossing number of P n 2C 3 ’, Discrete Math. 312 (2012), 20962101.
[17] Klešč, M., Petrillová, J. and Valo, M., ‘On the crossing numbers of cartesian products of wheels and trees’, Discrete Math. Graph Theory 37(2) (2017), 399413.
[18] McQuillan, D., Pan, S. and Richter, R. B., ‘On the crossing number of K 13 ’, J. Combin. Theory Ser. B 115 (2015), 224235.
[19] Richter, R. B. and Thomassen, C., ‘Intersections of curve systems and the crossing number of C 5 × C 5 ’, Discrete Comput. Geom. 13(1) (1995), 149159.
[20] Ringeisen, R. D. and Beineke, L. W., ‘The crossing number of C 3 × C n ’, J. Combin. Theory Ser. B 24(2) (1978), 134136.
[21] Zheng, W., Lin, X., Yang, Y. and Deng, C., ‘On the crossing number of K m C n and K m, l P n ’, Discrete Appl. Math. 156 (2008), 18921907.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed