Geodesic
The Crossing Numbers of Products of Path with Graphs of Order Six
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 571-582.

Voir la notice de l'article provenant de la source Library of Science

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path P_n of length n, the crossing numbers of Cartesian products G □ P_n for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G □ P_n 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 □ P_n is 2(n − 1). In addition, the crossing numbers of G □ P_n for fourty graphs G on six vertices are collected.
Keywords: graph, drawing, crossing number, Cartesian product, path, tree 
@article{DMGT_2013_33_3_a6,
     author = {Kle\v{s}\v{c}, Mari\'an and Petrillov\'a, Jana},
     title = {The {Crossing} {Numbers} of {Products} of {Path} with {Graphs} of {Order} {Six}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {571--582},
     publisher = {mathdoc},
     volume = {33},
     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a6/}
}
TY  - JOUR
AU  - Klešč, Marián
AU  - Petrillová, Jana
TI  - The Crossing Numbers of Products of Path with Graphs of Order Six
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2013
SP  - 571
EP  - 582
VL  - 33
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a6/
LA  - en
ID  - DMGT_2013_33_3_a6
ER  - 
%0 Journal Article
%A Klešč, Marián
%A Petrillová, Jana
%T The Crossing Numbers of Products of Path with Graphs of Order Six
%J Discussiones Mathematicae. Graph Theory
%D 2013
%P 571-582
%V 33
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a6/
%G en
%F DMGT_2013_33_3_a6
Klešč, Marián; Petrillová, Jana. The Crossing Numbers of Products of Path with Graphs of Order Six. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 571-582. http://geodesic.mathdoc.fr/item/DMGT_2013_33_3_a6/

[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 Pěstováaní 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