The Minimality of the Georges-Kelmans Graph

In 1971, Tutte wrote in an article that it is tempting to conjecture that every 3connected 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.

[1]  G. Brinkmann.  A Practical Algorithm for the Computation of the Genus , 2020, ArXiv.

[2]  Brendan D. McKay,et al.  The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices , 1988, J. Comb. Theory, Ser. B.

[3]  A. K. Kel'mans.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: Cubic bipartite cyclic 4-connected graphs without Hamiltonian circuits , 1988 .

[4]  William McCuaig,et al.  Edge reductions in cyclically k-connected cubic graphs , 1992, J. Comb. Theory, Ser. B.

[5]  W. T. Tutte.  On the 2-factors of bicubic graphs , 1971, Discret. Math..

[6]  B. McKay,et al.  Fast generation of planar graphs , 2007 .

[7]  Bill Jackson,et al.  Longest cycles in 3-connected cubic graphs , 1986, J. Comb. Theory, Ser. B.

[8]  Hadrien Mélot,et al.  House of Graphs: A database of interesting graphs , 2012, Discret. Appl. Math..

[9]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[10]  Frank Harary,et al.  Graph Theory , 2016 .

[11]  Markus Meringer,et al.  Fast generation of regular graphs and construction of cages , 1999, J. Graph Theory.

[12]  Mark N. Ellingham,et al.  Non-hamiltonian 3-connected cubic bipartite graphs , 1983, J. Comb. Theory, Ser. B.

[13]  J. D. Horton.  On two-factors of bipartite regular graphs , 1982, Discret. Math..

[14]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[15]  Kolja B. Knauer,et al.  Cuts in matchings of 3-connected cubic graphs , 2017, Eur. J. Comb..

[16]  Brendan D. McKay,et al.  Hamiltonian cycles in cubic 3-connected bipartite planar graphs , 1985, J. Comb. Theory, Ser. B.

[17]  W. T. Tutte.  On Hamiltonian Circuits , 1946 .

[18]  N. Wormald.  Models of random regular graphs , 2010 .

[19]  John P. Georges.  Non-hamiltonian bicubic graphs , 1989, J. Comb. Theory, Ser. B.

[20]  G. Brinkmann.  Fast generation of cubic graphs , 1996 .