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2013 | 33 | 3 | 583-597
Tytuł artykułu

On the Crossing Numbers of Cartesian Products of Stars and Graphs of Order Six

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. According to their special structure, the class of Cartesian products of two graphs is one of few graph classes for which some exact values of crossing numbers were obtained. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. Moreover, except of six graphs, the crossing numbers of Cartesian products G⃞K1,n for all other connected graphs G on five vertices are known. In this paper we are dealing with the Cartesian products of stars with graphs on six vertices. We give the exact values of crossing numbers for some of these graphs and we summarise all known results concerning crossing numbers of these graphs. Moreover, we give the crossing number of G1⃞T for the special graph G1 on six vertices and for any tree T with no vertex of degree two as well as the crossing number of K1,n⃞T for any tree T with maximum degree five.
Słowa kluczowe
Wydawca
Rocznik
Tom
33
Numer
3
Strony
583-597
Opis fizyczny
Daty
wydano
2013-07-01
online
2013-07-30
Twórcy
  • Faculty of Electrical Engineering and Informatics Technical University of Košice Letná 9, 042 00 Košice, Slovak Republic, marian.klesc@tuke.sk
  • Faculty of Electrical Engineering and Informatics Technical University of Košice Letná 9, 042 00 Košice, Slovak Republic, stefan.schrotter@tuke.sk
Bibliografia
  • 1] K. Asano, The crossing number of K1,3,n and K2,3,n, J. Graph Theory 10 (1986) 1-8. doi:10.1002/jgt.3190100102[Crossref]
  • [2] 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[Crossref]
  • [3] 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[Crossref]
  • [4] D. Bokal, On the crossing numbers of Cartesian products with trees, J. Graph Theory 56 (2007) 287-300. doi:10.1002/jgt.20258[Crossref]
  • [5] M. Draženská and M. Klešč, The crossing numbers of products of the graph K2,2,2 with stars, Carpathian J. Math. 24 (2008) 327-331.
  • [6] L.Y. Glebsky and G. Salazar, The crossing number of Cm ×Cn is as conjectured for n ≥ m(m + 1), J. Graph Theory 47 (2004) 53-72. doi:10.1002/jgt.20016[Crossref]
  • [7] F. Harary, P.C. Kainen and A.J. Schwenk, Toroidal graphs with arbitrarily high crossing numbers, Nanta Math 6 (1973) 58-67.
  • [8] X. He, The crossing number of Cartesian products of stars with 5-vertex graphs, in: 2010 International Conference on Computational Intelligence and Software Engineering, CiSE 2010, Wuhan, December 2010.
  • [9] P.T. Ho, The crossing number of K2,2,2,n, Far East J. Appl. Math. 30 (2008) 43-69.
  • [10] Y. Huang and T. Zhao, The crossing number of K1,4,n, Discrete Math. 308 (2008) 1634-1638. doi:10.1016/j.disc.2006.12.002[Crossref]
  • [11] S. Jendrol’ and M. Ščerbová, On the crossing numbers of Sm × Pn and Sm × Cn, ˇ Casopis pro P ˇ estov´ an´ı Matematiky 107 ( 1982) 225-230.
  • [12] D.J. Kleitman, The crossing number of K5,n, J. Combin. Theory (B) 9 (1971) 315-323.
  • [13] M. Klešč, The crossing numbers of Cartesian products of stars and paths or cycles, Math. Slovaca 41 (1991) 113-120.
  • [14] M. Klešč, The crossing numbers of products of paths and stars with 4-vertex graphs, J. Graph Theory 18 (1994) 605-614.
  • [15] M. Klešč, The crossing number of K2,3 ×Pn and K2,3 ×Sn, Tatra Mt. Math. Publ. 9 (1996) 51-56.
  • [16] M. Klešč, On the crossing numbers of products of stars and graphs of order five, Graphs Combin. 17 (2001) 289-294. doi:10.1007/s003730170042[Crossref]
  • [17] M. Klešč, The join of graphs and crossing numbers, Electron. Notes Discrete Math. 28 (2007) 349-355. doi:10.1016/j.endm.2007.01.049[Crossref]
  • [18] M. Klešč, On the crossing numbers of Cartesian products of stars and graphs on five vertices, Combinatorial Algorithms, Springer, LNCS 5874 (2009) 324-333. doi:10.1007/978-3-642-10217-2 32[Crossref]
  • [19] V.R. Kulli and M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97.
  • [20] S. Lü and Y. Huang, On the crossing numner of K5 × Sn, J. Math. Res. Expo. 28 (2008) 445-459.
  • [21] H. Mei and Y. Huang, The crossing number of K1,5,n, Internat. J. Math. Combin. 1 (2007) 33-44.
  • [22] K. Zarankiewicz, On a problem of P. Turán concerning graphs, Fund. Math 41 (1954) 137-145
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1705
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