Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-xbbwl Total loading time: 1.298 Render date: 2021-02-26T02:44:16.572Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

MICHAEL HAYTHORPE
Affiliation:
1284 South Road, Clovelly Park, 5042, College of Science and Engineering, Flinders University, Australia email michael.haythorpe@flinders.edu.au
ALEX NEWCOMBE
Affiliation:
1284 South Road, Clovelly Park, 5042, College of Science and Engineering, Flinders University, Australia email alex.newcombe@flinders.edu.au

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.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Access options

Get access to the full version of this content by using one of the access options below.

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), 97107.Google Scholar
Anderson, M., Richter, R. B. and Rodney, P., ‘The crossing number of C 7 × C 7 ’, Congr. Numer. 125 (1996), 97117.Google Scholar
Asano, K., ‘The crossing number of K 1, 3, n and K 2, 3, n ’, J. Graph Theory 10 (1986), 18.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), 145155.CrossRefGoogle Scholar
Bokal, D., ‘On the crossing number of Cartesian products with paths’, J. Combin. Theory Ser. B 97(3) (2007), 381384.CrossRefGoogle 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), 18031829.CrossRefGoogle 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), 29.129.13.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), 125129.CrossRefGoogle Scholar
Garey, M. R. and Johnson, D. S., ‘Crossing number is NP-complete’, SIAM J. Algebr. Discrete Methods 4(3) (1983), 312316.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Harary, F., Kainen, P. C. and Schwenk, A. J., ‘Toroidal graphs with arbitrarily high crossing numbers’, Nanta Math. 6(1) (1973), 5867.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), 225230.Google Scholar
Klešč, M., ‘On the crossing numbers of Cartesian products of stars and paths or cycles’, Math. Slovaca 41(2) (1991), 113120.Google Scholar
Klešč, M. and Kravecová, D., ‘The crossing number of P n 2C 3 ’, Discrete Math. 312 (2012), 20962101.CrossRefGoogle 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), 399413.CrossRefGoogle Scholar
McQuillan, D., Pan, S. and Richter, R. B., ‘On the crossing number of K 13 ’, J. Combin. Theory Ser. B 115 (2015), 224235.CrossRefGoogle 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), 149159.CrossRefGoogle Scholar
Ringeisen, R. D. and Beineke, L. W., ‘The crossing number of C 3 × C n ’, J. Combin. Theory Ser. B 24(2) (1978), 134136.CrossRefGoogle 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), 18921907.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 66 *
View data table for this chart

* Views captured on Cambridge Core between 08th February 2019 - 26th February 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *