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ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH
Published online by Cambridge University Press: 08 February 2019
Abstract
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
References
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 , eurocg15.fri.uni-lj.si/pub/eurocg15-book-of-abstracts.pdf, 57–60.Google Scholar
Anderson, M., Richter, R. B. and Rodney, P., ‘The crossing number of C
6 × C
6
’, Congr. Numer.
117 (1996), 97–107.Google Scholar
Anderson, M., Richter, R. B. and Rodney, P., ‘The crossing number of C
7 × C
7
’, Congr. Numer.
125 (1996), 97–117.Google Scholar
Asano, K., ‘The crossing number of K
1, 3, n
and K
2, 3, n
’, J. Graph Theory
10 (1986), 1–8.Google Scholar
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), 145–155.Google Scholar
Bokal, D., ‘On the crossing number of Cartesian products with paths’, J. Combin. Theory Ser. B
97(3) (2007), 381–384.Google Scholar
Cabello, S. and Mohar, B., ‘Adding one edge to planar graphs makes crossing number and 1-planarity hard’, SIAM J. Comput.
42(5) (2013), 1803–1829.Google Scholar
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), 29.1–29.13.Google Scholar
Clancy, K., Haythorpe, M. and Newcombe, A., ‘An effective crossing minimisation heuristic based on star insertion’, preprint available at arXiv:abs/1804.09900.Google Scholar
Dean, A. M. and Richter, R. B., ‘The crossing number of C
4 × C
4
’, J. Graph Theory
19(1) (1995), 125–129.Google Scholar
Garey, M. R. and Johnson, D. S., ‘Crossing number is NP-complete’, SIAM J. Algebr. Discrete Methods
4(3) (1983), 312–316.Google Scholar
Glebsky, L. Y. and Salazar, G., ‘The crossing number of C
m
× C
n
is as conjectured for n ≥ m (m + 1)’, J. Graph Theory
47(1) (2004), 53–72.Google Scholar
Harary, F., Kainen, P. C. and Schwenk, A. J., ‘Toroidal graphs with arbitrarily high crossing numbers’, Nanta Math.
6(1) (1973), 58–67.Google Scholar
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), 225–230.Google Scholar
Klešč, M., ‘On the crossing numbers of Cartesian products of stars and paths or cycles’, Math. Slovaca
41(2) (1991), 113–120.Google Scholar
Klešč, M. and Kravecová, D., ‘The crossing number of P
n
2◻C
3
’, Discrete Math.
312 (2012), 2096–2101.Google Scholar
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), 399–413.Google Scholar
McQuillan, D., Pan, S. and Richter, R. B., ‘On the crossing number of K
13
’, J. Combin. Theory Ser. B
115 (2015), 224–235.Google Scholar
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), 149–159.Google Scholar
Ringeisen, R. D. and Beineke, L. W., ‘The crossing number of C
3 × C
n
’, J. Combin. Theory Ser. B
24(2) (1978), 134–136.Google Scholar
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), 1892–1907.Google Scholar