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)


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.


Variable neighborhood search Uniquely Hamiltonian graphs Combinatorial optimization 


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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

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