## Nonhamiltonian 3-connected cubic planar graphs

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Venue: | SIAM J. Disc. Math |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Aldred_nonhamiltonian3-connected,

author = {R. E. L. Aldred and S. Bau and D. A. Holton and Brendan D. Mckay},

title = {Nonhamiltonian 3-connected cubic planar graphs},

journal = {SIAM J. Disc. Math},

year = {},

volume = {13},

pages = {2000}

}

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### Abstract

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. That all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified. 1 Introduction. In this paper we describe an investigation (making much use of computation) of cyclically k-connected cubic planar graphs (CkCPs) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CPs. The investigation

### Citations

275 | Convex Polytopes
- Grünbaum
- 2003
(Show Context)
Citation Context ...ial 4-cut (i.e. the only non-trivial 4-cuts in G are the cocyles of 4-cycles and there must be at least one such 4-cut) and |V (G)| ∈{38, 40, 42}. (The graph NH42.a in Figure 2 above is from Grünbaum =-=[8]-=- and is a nonhamiltonian C4CP on 42 vertices with no essential 4-cuts.) 2 The cyclically 4-connected case. Throughout this section we shall assume that G is a nonhamiltonian C4CP of smallest order, ot... |

177 |
nauty user’s guide (version 1.5
- McKay
- 1990
(Show Context)
Citation Context ...nnected cubic planar graphs on up to 32 vertices which contain a b-edge but no triangle. The method used was that of Mohar [13], in conjunction with the graph isomorphism system described by McKay in =-=[11]-=-. Phase 2. Note that a C3CP H with a triangle has a b-edge if and only if the C3CP obtained from H by contracting the triangle has a b-edge. Thus the C3CPs on up to 32 vertices, with b-edges and trian... |

15 |
The smallest non-Hamiltonian 3-connected cubic planar graphs have 38 vertices
- Holton, McKay
- 1988
(Show Context)
Citation Context ... shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CPs. The investigation extends the work of Holton and McKay in =-=[10]-=- in which the smallest order of a nonhamiltonian C3CP was shown to be 38. We provide answers to two questions raised in that paper, namely: (a) What is the smallest order of a nonhamiltonian C4CP? (b)... |

7 |
On generating planar graphs
- Barnette
- 1974
(Show Context)
Citation Context ...h that the graph known 5sto exhibit this bound, NH44 of Figure 4, could not be guaranteed to be unique. We have now generated all C5CPs on up to 52 vertices using the generation procedure of Barnette =-=[1]-=- and Butler [6]. The graphs were then checked for hamiltonian cycles. This search confirms that NH44 is the unique nonhamiltonian C5CP of smallest order. It also yielded a complete list of all nonhami... |

7 |
A generation procedure for the simple 3-polytopes with cyclically 5-connected graphs
- Butler
- 1974
(Show Context)
Citation Context ...h known 5sto exhibit this bound, NH44 of Figure 4, could not be guaranteed to be unique. We have now generated all C5CPs on up to 52 vertices using the generation procedure of Barnette [1] and Butler =-=[6]-=-. The graphs were then checked for hamiltonian cycles. This search confirms that NH44 is the unique nonhamiltonian C5CP of smallest order. It also yielded a complete list of all nonhamiltonian C5CPs o... |

4 | Cycles in 3-connected cubic planar graphs - Holton, McKay |

3 |
PentHex Puzzles -- a reliable and efficient top-down approach to fullerene-structure enumeration
- Brinkmann, Dress
(Show Context)
Citation Context ...tested them for hamiltonicity. All were found to be hamiltonian. This supports the conjecture and extends the known hamiltonicity of fullerenes beyond 150 vertices (established in Brinkmann and Dress =-=[3]-=-). 5 Some graph counts. In the table below we include some numbers of the graphs produced in our computations. The numbers in the column headed n1 are approximate total numbers of C3CPs of each of the... |

2 | Hamiltonian lines in cubic graphs - Bosák - 1966 |

2 |
Non-hamiltonian simple planar maps
- Faulkner, Younger
- 1974
(Show Context)
Citation Context ...ng result. Theorem 4 All C4CPs on at most 40 vertices are hamiltonian. Furthermore, the only nonhamiltonian C4CPs on 42 vertices are the three shown in Figure 2. 3 The cyclically 5-connected case. In =-=[7]-=-, Faulkner and Younger established that the smallest nonhamiltonian C5CP has 44 vertices. The nature of the search employed there was such that the graph known 5sto exhibit this bound, NH44 of Figure ... |

1 | A constructive enumeration of fullerenes, submitted - Brinkmann, Dress |

1 |
Fast generation of 3-connected cubic planar graphs, in preparation
- Brinkmann, McKay
(Show Context)
Citation Context ...e size at most 6 are hamiltonian. In particular, this conjecture covers the fullerenes, C3CPs with 12 faces of size 5 and all other faces of size 6. Using generation techniques of Brinkmann and McKay =-=[5]-=- we generated all C3CPs with maximum face size at most 6 (note that all such graphs on more than 20 vertices must have faces of size 6) up to 176 vertices and tested them for hamiltonicity. All were f... |

1 |
A fast search algorithm for hamiltonian cycles in cubic graphs
- McKay
(Show Context)
Citation Context ...orbidden cuts. Phase 4. Test the graphs yielded by Phase 3 for hamiltonicity and discard those with hamilton cycles. (All hamiltonicity checking in our computations was done using the method of McKay =-=[12]-=-.) No graphs remained at the completion of Phase 4, proving the following theorem. Theorem 3 Let G be a nonhamiltonian C4CP of order 38, 40 or 42. Then G contains none of the subgraphs in Figure 3. Ph... |

1 |
Search for minimal non-hamiltonian simple 3-polytopes
- Mohar
- 1983
(Show Context)
Citation Context ...a expansion. Our computation proceeded as follows. Phase 1. Generate all 3-connected cubic planar graphs on up to 32 vertices which contain a b-edge but no triangle. The method used was that of Mohar =-=[13]-=-, in conjunction with the graph isomorphism system described by McKay in [11]. Phase 2. Note that a C3CP H with a triangle has a b-edge if and only if the C3CP obtained from H by contracting the trian... |

1 |
Every simple 3-polytope of order 32 or less is hamiltonian
- Okamura
- 1982
(Show Context)
Citation Context ...ssential 4-cuts, triangles and adjacent 4-cycles are forbidden subgraphs. Similarly, cuts of sizes 1 and 2, non-trivial 3-cuts and essential 4-cuts are forbidden cuts. Following the method of Okamura =-=[14]-=-, we wish to show that none of the four subgraphs in Figure 3 can occur in G. For each subgraph H in Figure 3, the reduction 3sH(G) ofG using H is obtained by successive edge reduction (i.e. deleting ... |