The minimality of the Georges–Kelmans graph
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- by Gunnar Brinkmann, Jan Goedgebeur and Brendan D. McKay;
- Math. Comp. 91 (2022), 1483-1500
- DOI: https://doi.org/10.1090/mcom/3701
- Published electronically: November 17, 2021
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Abstract:
In 1971, Tutte wrote in an article that it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian. Motivated by this remark, Horton constructed a counterexample on $96$ vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts—even if they have girth 6—it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs.
In 1969 Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette’s conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.
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Bibliographic Information
- Gunnar Brinkmann
- Affiliation: Department of Applied Mathematics, Computer Science & Statistics, Ghent University, 9000 Ghent, Belgium
- MR Author ID: 252206
- Email: gunnar.brinkmann@ugent.be
- Jan Goedgebeur
- Affiliation: Department of Computer Science, KU Leuven Campus Kulak, 8500 Kortrijk, Belgium; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, 9000 Ghent, Belgium; and Computer Science Department, University of Mons, 7000 Mons, Belgium
- MR Author ID: 945364
- ORCID: 0000-0001-8984-2463
- Email: jan.goedgebeur@kuleuven.be
- Brendan D. McKay
- Affiliation: Department of Computing, Australian National University, Canberra Australian Capitol Territory 2601, Australia
- MR Author ID: 122480
- Email: brendan.mckay@anu.edu.au
- Received by editor(s): March 17, 2021
- Received by editor(s) in revised form: August 31, 2021
- Published electronically: November 17, 2021
- Additional Notes: The second author was supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). The third author was supported by a Francqui International Professorship.
- © Copyright 2021 by the authors
- Journal: Math. Comp. 91 (2022), 1483-1500
- MSC (2020): Primary 05C45, 05C85, 05C10
- DOI: https://doi.org/10.1090/mcom/3701
- MathSciNet review: 4405503