Advertisement

Finding Uniquely Hamiltonian Graphs of Minimum Degree Three with Small Crossing Numbers

  • Benedikt KlockerEmail author
  • Herbert Fleischner
  • Günther R. Raidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9668)

Abstract

In graph theory, a prominent conjecture of Bondy and Jackson states that every uniquely hamiltonian planar graph must have a vertex of degree two. In this work we try to find uniquely hamiltonian graphs with minimum degree three and a small crossing number by minimizing the number of crossings in an embedding and the number of degree-two vertices. We formalize an optimization problem for this purpose and propose a general variable neighborhood search (GVNS) for solving it heuristically. The several different types of used neighborhoods also include an exponentially large neighborhood that is effectively searched by means of branch and bound. To check feasibility of neighbors we need to solve hamiltonian cycle problems, which is done in a delayed manner to minimize the computation effort. We compare three different configurations of the GVNS. Although our implementation could not find a uniquely hamiltonian planar graph with minimum degree three disproving Bondy and Jackson’s conjecture, we were able to find uniquely hamiltonian graphs of minimum degree three with crossing number four for all number of vertices from 10 to 100.

Keywords

Variable neighborhood search Uniquely Hamiltonian graphs Combinatorial optimization 

References

  1. 1.
    Abbasi, S., Jamshed, A.: A degree constraint for uniquely Hamiltonian graphs. Graphs and Combinatorics 22(4), 433–442 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  3. 3.
    Bevc, S., Savnik, I.: Using tries for subset and superset queries. In: Proceedings of the ITI 2009, pp. 147–152 (2009)Google Scholar
  4. 4.
    Bondy, J.A., Jackson, B.: Vertices of small degree in uniquely Hamiltonian Graphs. J. Comb. Theory Ser. B 74(2), 265–275 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caporossi, G., Hansen, P.: Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system. Discrete Math. 212(1–2), 29–44 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Charikar, M., Indyk, P., Panigrahy, R.: New algorithms for subset query, partial match, orthogonal range searching, and related problems. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 451–462. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Cook, W.: Concorde TSP Solver (2011). http://www.math.uwaterloo.ca/tsp/concorde/. Accessed on 31 Jan 2016
  8. 8.
    Entringer, R.C., Swart, H.: Spanning cycles of nearly cubic graphs. J. Comb. Theory Ser. B 29(3), 303–309 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fleischner, H.: Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs. J. Graph Theory 18(5), 449–459 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fleischner, H.: Uniquely Hamiltonian graphs of minimum degree 4. J. Graph Theory 75(2), 167–177 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garey, M., Johnson, D.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gould, R.J.: Advances on the Hamiltonian problem-a survey. Graphs and Combinatorics 19(1), 7–52 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hansen, P., Mladenović, N.: An introduction to variable neighborhood search. In: Voss, S., et al. (eds.) Metaheuristics, Advances and Trends in Local Search Paradigms for Optimization, pp. 433–458. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  14. 14.
    Hansen, P., Mladenović, N.: A tutorial on variable neighborhood search. Technical report G-2003-46, GERAD, July 2003Google Scholar
  15. 15.
    Helmer, S., Aly, R., Neumann, T., Moerkotte, G.: Indexing set-valued attributes with a multi-level extendible hashing scheme. In: Wagner, R., Revell, N., Pernul, G. (eds.) DEXA 2007. LNCS, vol. 4653, pp. 98–108. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Helsgaun, K.: Effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Helsgaun, K.: LKH (2012). http://www.akira.ruc.dk/~keld/research/LKH/. Accessed 03 Feb 2016
  18. 18.
    Jayram, T.S., Khot, S., Kumar, R., Rabani, Y.: Cell-probe lower bounds for the partial match problem. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC 2003, pp. 667–672. ACM, New York, (2003)Google Scholar
  19. 19.
    Klocker, B., Raidl, G.: Finding uniquely hamiltonian graphs with minimal degree three. Technical report, Algorithms and Complexity Group, TU Wien (2016)Google Scholar
  20. 20.
    Pisinger, D., Ropke, S.: Large neighborhood search. In: Gendreau, M., Potvin, J.-Y. (eds.) Handbook of Metaheuristics, pp. 399–419. Springer US, London (2010)CrossRefGoogle Scholar
  21. 21.
    Sheehan, J.: The multiplicity of Hamiltonian circuits in a graph. In: Recent Advances in Graph Theory, pp. 477–480 (1975)Google Scholar
  22. 22.
    Thomason, A.G.: Hamiltonian cycles and uniquely edge colourable graphs. Ann. Discrete Math. 3, 259–268 (1978). Advances in Graph TheoryMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Benedikt Klocker
    • 1
    Email author
  • Herbert Fleischner
    • 1
  • Günther R. Raidl
    • 1
  1. 1.Institute of Computer Graphics and AlgorithmsTU WienViennaAustria

Personalised recommendations