Home The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
Article
Licensed
Unlicensed Requires Authentication

The influence of separating cycles in drawings of K5e in the join product with paths and cycles

  • Michal Staš EMAIL logo and Mária Timková
Published/Copyright: October 15, 2024
Become an author with De Gruyter Brill

Abstract

The crossing number cr(H) of a graph H is the minimum number of edge crossings over all drawings of H in the plane. Let H be the connected graph of order five isomorphic to K5e obtained by removing one edge from the complete graph K5. The main aim of the paper is to give the crossing numbers of the join products H + Pn and H + Cn, where Pn and Cn are the path and the cycle on n vertices, respectively. The proofs are done with the help of a suitable classification of a large number of drawings of the graph H in view of the existence of a separating cycle of two possible types.

  1. Communicated by Peter Horák

References

[1] Asano, K.: The crossing number of K1,3,n and K2,3,n, J. Graph Theory 10(1) (1986), 1–8.10.1002/jgt.3190100102Search in Google Scholar

[2] Clancy, K.—Haythorpe, M.—Newcombe, A.: A survey of graphs with known or bounded crossing numbers, Australas. J. Combin. 78(2) (2020), 209–296.Search in Google Scholar

[3] Chimani, M.—Wiedera, T.: An ILP-based proof system for the crossing number problem, 24th In Proceedings of the Annual European Symposium on Algorithms (ESA 2016) 29 (2016), 1–13.Search in Google Scholar

[4] Fernau, H.—Fomin, F. V.—Lokshtanov, D.—Mnich, M.—Philip, G.—Saurabh, S.: Social choice meets graph drawing: How to get subexponential time algorithms for ranking and drawing problems, Tsinghua Sci. Technol. 19 (2014), 374–386.10.1109/TST.2014.6867519Search in Google Scholar

[5] Fridman G.—Vasiliev, Y.—Puhkalo, V.—Ryzhov, V.: A mixed-integer program for drawing orthogonal hyperedges in a hierarchical hypergraph, Mathematics 10(5) (2022), 689.10.3390/math10050689Search in Google Scholar

[6] Garey, M. R.—Johnson, D. S.: Crossing number is NP-complete SIAM J. Algebraic. Discrete Methods, 4(3) (1983), 312–316.10.1137/0604033Search in Google Scholar

[7] Gethner, E.—Hogben, L.—Lidicky, B.—Pfender, F.—Ruiz, A.—Young, M.: On crossing numbers of complete tripartite and balanced complete multipartite graphs, J. Graph Theory 84 (2017), 552–565.10.1002/jgt.22041Search in Google Scholar

[8] Hernández-Vélez, C.—Medina, C.—Salazar, G.: The optimal drawing of K5,n, Electron. J. Combin. 21(4) (2014), # P4.1.10.37236/2777Search in Google Scholar

[9] Ho, P. T.: On the crossing number of some complete multipartite graphs, Utilitas Math. 79 (2009), 125–143.Search in Google Scholar

[10] Huang, Y.—Zhao, T.: The crossing number of K1,4,n, Discrete Math. 308(9) (2008), 1634–1638.10.1016/j.disc.2006.12.002Search in Google Scholar

[11] Jenny, B.—Stephen, D. M.—Muehlenhaus, I.—Marston, B. E.—Sharma, R.—Zhang, E.—Jenny, H.: Design principles for origin-destination flow maps, Cartogr. Geogr. Inf. Sci. 45 (2018), 62–75.10.1080/15230406.2016.1262280Search in Google Scholar

[12] Kleitman, D. J.: The crossing number of K5,n, J. Combin. Theory 9 (1970), 315–323.10.1016/S0021-9800(70)80087-4Search in Google Scholar

[13] Klešč, M.: The crossing number of join of the special graph on six vertices with path and cycle, Discrete Math. 310(9) (2010), 1475–1481.10.1016/j.disc.2009.08.018Search in Google Scholar

[14] Klešč, M.: The join of graphs and crossing numbers, Electron. Notes in Discrete Math. 28 (2007), 349–355.10.1016/j.endm.2007.01.049Search in Google Scholar

[15] Klešč, M.: The crossing numbers of join of cycles with graphs of order four, Proc. Aplimat 2019: 18th Conference on Applied Mathematics (2019), 634–641.Search in Google Scholar

[16] Klešč, M.: On the crossing numbers of products of stars and graphs of order five, Graphs Comb. 17(2) (2001), 289–294.10.1007/s003730170042Search in Google Scholar

[17] Klešč, M.—Petrillová, J.—Valo, M.: On the crossing numbers of Cartesian products of wheels and trees, Discuss. Math. Graph Theory 37(2) (2017), 399–413.10.7151/dmgt.1957Search in Google Scholar

[18] Klešč, M.—Schrötter, Š.: The crossing numbers of join products of paths with graphs of order four, Discuss. Math. Graph Theory 31(2) (2011), 321–331.10.7151/dmgt.1548Search in Google Scholar

[19] Klešč, M.—Schrötter, Š.: The crossing numbers of join of paths and cycles with two graphs of order five, Combinatorial Algorithms, Springer, LNCS 7125 (2012), 160–167.10.1007/978-3-642-28212-6_15Search in Google Scholar

[20] Klešč, M.—Staš, M.: Cyclic permutations in determining crossing numbers, Discuss. Math. Graph Theory 42(4) (2022), 1163–1183.10.7151/dmgt.2351Search in Google Scholar

[21] Lü, S.—Huang, Y.: The crossing number of K5 × Sn, J. Math. Res. Exp. 28(3) (2008), 445–459.Search in Google Scholar

[22] Nath, R. K.—Sen, B.—Sikdar, B. K.: Optimal synthesis of QCA logic circuit eliminating wire-crossings, IET Circuits Devices Syst. 11 (2017), 201–208.10.1049/iet-cds.2016.0252Search in Google Scholar

[23] Schaefer, M.: The Graph Crossing Number and its Variants: A Survey, Electron. J. Combin. (2013)10.37236/2713Search in Google Scholar

[24] Staš, M.: On the crossing number of join product of the discrete graph with special graphs of order five, Electron. J. Graph Theory Appl. 8(2) (2020), 339–351.10.5614/ejgta.2020.8.2.10Search in Google Scholar

[25] Staš, M.: Join Products K2,3 + Cn, Mathematics 8(6) (2020), Art. No. 925.10.3390/math8060925Search in Google Scholar

[26] Staš, M.: On the crossing numbers of join K2,3 + Pn, Proc. Aplimat 2020: 19th Conference on Applied Mathematics (2020), 991–998.Search in Google Scholar

[27] Staš, M.: The crossing numbers of join products of paths and cycles with four graphs of order five, Mathematics 9(11) (2021), Art. No. 1277.10.3390/math9111277Search in Google Scholar

[28] Staš, M.—Švecová, M.: The crossing numbers of join products of paths with three graphs of order five, Opuscula Math. 42(4) (2022), 635–651.10.7494/OpMath.2022.42.4.635Search in Google Scholar

[29] Staš, M.—Timková, M.: The crossing numbers of join products of four graphs of order five with paths and cycles, Opuscula Math. 43(6) (2023), 865–883.10.7494/OpMath.2023.43.6.865Search in Google Scholar

[30] Staš, M.—Valiska, J.: On the crossing numbers of join products of W4 + Pn and W4 + Cn, Opuscula Math. 41(1) (2021), 95–112.10.7494/OpMath.2021.41.1.95Search in Google Scholar

[31] Su, Z.: The Crossing number of the join product of K1,1,1,2 + Pn, Acta Math. Appl. Sinica 40(3) (2017), 345–354.Search in Google Scholar

[32] Woodall, D. R.: Cyclic-order graphs and Zarankiewicz’s crossing number conjecture, J. Graph Theory 17(6) (1993), 657–671.10.1002/jgt.3190170602Search in Google Scholar

Received: 2023-10-26
Accepted: 2024-03-15
Published Online: 2024-10-15
Published in Print: 2024-10-28

© 2024 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. On monoids of endomorphisms of a cycle graph
  2. The influence of separating cycles in drawings of K5e in the join product with paths and cycles
  3. Completely hereditarily atomic OMLS
  4. New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
  5. On the monogenity and Galois group of certain classes of polynomials
  6. Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
  7. Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
  8. Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
  9. Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
  10. The discrete fractional Karamata theorem and its applications
  11. Complex Generalized Stancu-Schurer Operators
  12. New oscillatory criteria for third-order differential equations with mixed argument
  13. Cauchy problem for a loaded hyperbolic equation with the Bessel operator
  14. Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
  15. On the rate of convergence for the q-Durrmeyer polynomials in complex domains
  16. Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
  17. On the relation of Kannan contraction and Banach contraction
  18. Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
  19. Residual Inaccuracy Extropy and its properties
  20. Archimedean -groups with strong unit: cozero-sets and coincidence of types of ideals
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0079/html
Scroll to top button