Hypohamiltonian planar cubic graphs with girth five

Hypohamiltonian planar cubic graphs with girth five

Brendan D. McKay
Research School of Computer Science
Australian National University
ACT 2601, Australia
E-mail: bdm@cs.anu.edu.au
Abstract

A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity five, and thus girth five. We show by computer search that the smallest members of this class are three graphs with 76 vertices.

Keywords: graph generation; hypohamiltonian graph; planar graph; cubic graph

MSC 2010. 05C10, 05C30, 05C38, 05C45, 05C85.

1 Introduction

A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Such graphs exist on orders 10 (the Petersen graph), 13, 15, 16, and all orders from 18 onwards [1]. It is elementary to show that hypohamiltonian graphs must be 3-connected and have cyclic connectivity at least 4.

A substantial body of literature is devoted to finding hypohamiltonian graphs with various properties. The first planar hypohamiltonian graph, with 105 vertices, was found by Thomassen [8]; the present smallest known order is 40 vertices but smaller orders have not been ruled out [7].

Planar hypohamiltonian graphs can be cubic, as first shown by Thomassen with an example on 94 vertices [9]. The smallest examples found so far have 70 vertices; the first example by Araya and Wiener [2] and six more by Jooyandeh and McKay [6].

Planar hypohamiltonian graphs can also have girth 5, and in this case the smallest order is proven to be 45 [7]. However, it was not known until now whether planar hypohamiltonian graphs can be both cubic and of girth 5. Our purpose is to present the first examples.

2 The results

Using the program plantri [4, 5] we generated all planar cubic graphs of girth 5 and cyclic connectivity at least 4 on up to 76 vertices. No theory is known which could efficiently restrict the search to a small subclass sure to contain the hypohamiltonian graphs, so we just tested all of them.

The computational task was daunting as more than graphs are involved and 76 vertices is large enough that a primitive backtrack search for hamiltonian cycles takes up to several hours per graph. Fortunately we have a program (unpublished) specifically designed for sub-cubic graphs that takes only 10 microseconds on average to find a hamiltonian cycle if there is one, and 15 milliseconds on average to rigorously prove that there is none. Even then the task took about 8 years of cpu time.

The results are shown in Table 1. For at most 74 vertices there are no hypohamiltonian graphs, but for 76 vertices there are three. All of them have cyclic connectivity 5. The graphs themselves are shown in Figures 13, which were drawn by CaGe [3].

Note that the graphs have different face counts despite having some similarities. The non-hamiltonicity of these graphs does not follow immediately from Grinberg’s condition, so we tested them with several independent programs to remove the possibility of error. Contrary to our usual experience of extremal graphs, none of them have any non-trivial automorphisms. We did not find a way to generalize these examples to larger sizes despite trying several approaches, so the problem of finding an infinite family remains open.

Figure 1: Hypohamiltonian graph with face count
Figure 2: Hypohamiltonian graph with face count
Figure 3: Hypohamiltonian graph with face count
20 1 1
22 0 0
24 1 1
26 1 1
28 3 3
30 4 4
32 12 12
34 23 23
36 2 71
38 4 187
40 22 627
42 84 1970
44 376 1 6833 1
46 1579 3 23384 1
48 6751 1 82625 0
50 27969 3 292164 3
52 115423 6 1045329 6
54 467948 12 3750277 2
56 1882184 49 13532724 22
58 7496828 126 48977625 37
60 29667311 214 177919099 31
62 116710547 659 648145255 194
64 457122502 1467 2368046117 298
66 1783850057 3247 8674199554 306
68 6941579864 9187 31854078139 1538
70 26950926431 22069 117252592450 2566
72 104455609591 50514 432576302286 3091
74 404298188921 137787 1599320144703 13487
76 1563255455769 339804 5925181102878 22274 3

The numbers and first appeared in [4].

Table 1: Counts of planar cyclically 4-connected cubic graphs of girth 5

References

  • [1] R. E. L. Aldred, B. D. McKay, and N. C. Wormald, Small hypohamiltonian graphs, J. Combin. Math. Combin. Comput. 23 (1997) 143–152.
  • [2] M. Araya and G. Wiener, On cubic planar hypohamiltonian and hypotraceable graphs, Electron. J. Combin. 18(1) (2011) #P85.
  • [3] G. Brinkmann, O. Delgado Friedrichs, S. Lisken, A. Peeters and N. Van Cleemput, CaGe—a virtual environment for studying some special classes of plane graphs—an update, MATCH Commun. Math. Comput. Chem. 63 (2010) 533–552.
  • [4] G. Brinkmann and B. D. McKay, Construction of planar triangulations with minimum degree 5, Discrete Math. 301 (2005) 147–163.
  • [5] G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem. 58 (2007) 323–357.
  • [6] M. Jooyandeh and B. D. McKay, Hypohamiltonian planar graphs, Web site at http://cs.anu.edu.au/bdm/data/planegraphs.html.
  • [7] M. Jooyandeh, B. D. McKay, P. Östergård, V. Pettersson and C. T. Zamfirescu, Planar hypohamiltonian graphs on 40 vertices, submitted (2013). arXiv:1302.2698
  • [8] C. Thomassen, Planar and infinite hypohamiltonian and hypotraceable graphs, Discrete Math. 14 (1976) 377–389.
  • [9] C. Thomassen, Planar cubic hypohamiltonian and hypotraceable graphs, J. Combin. Theory, Ser. B 30 (1981) 36–44.
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