### Abstract

We show that the probability of a 3-connected cubic map with 2n vertices being hamiltonian tends to zero exponentially with with n. We show that if there is one 3-connected triangulation which is not 4-colourable then the probability that a 3-connected triangulation is 4-colourable tends to zero exponentially with n. These results both follow easily from the result proved here that any given 3-connected triangulation, T, is contained (with the boundary of T an interior 3-cycle) in a 3-connected triangulation with 2_{n} faces with probability 1 + 0(c^{n}), c < 1.

Original language | English |
---|---|

Pages (from-to) | 141-149 |

Number of pages | 9 |

Journal | North-Holland Mathematics Studies |

Volume | 115 |

Issue number | C |

DOIs | |

Publication status | Published - 1 Jan 1985 |

Externally published | Yes |

### Cite this

*North-Holland Mathematics Studies*,

*115*(C), 141-149. https://doi.org/10.1016/S0304-0208(08)73003-2

}

*North-Holland Mathematics Studies*, vol. 115, no. C, pp. 141-149. https://doi.org/10.1016/S0304-0208(08)73003-2

**On Hamilton Cycles in 3-Connected Cubic Maps.** / Richmond, L. Bruce; Robinson, R. W.; Wormald, N. C.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - On Hamilton Cycles in 3-Connected Cubic Maps

AU - Richmond, L. Bruce

AU - Robinson, R. W.

AU - Wormald, N. C.

PY - 1985/1/1

Y1 - 1985/1/1

N2 - We show that the probability of a 3-connected cubic map with 2n vertices being hamiltonian tends to zero exponentially with with n. We show that if there is one 3-connected triangulation which is not 4-colourable then the probability that a 3-connected triangulation is 4-colourable tends to zero exponentially with n. These results both follow easily from the result proved here that any given 3-connected triangulation, T, is contained (with the boundary of T an interior 3-cycle) in a 3-connected triangulation with 2n faces with probability 1 + 0(cn), c < 1.

AB - We show that the probability of a 3-connected cubic map with 2n vertices being hamiltonian tends to zero exponentially with with n. We show that if there is one 3-connected triangulation which is not 4-colourable then the probability that a 3-connected triangulation is 4-colourable tends to zero exponentially with n. These results both follow easily from the result proved here that any given 3-connected triangulation, T, is contained (with the boundary of T an interior 3-cycle) in a 3-connected triangulation with 2n faces with probability 1 + 0(cn), c < 1.

UR - http://www.scopus.com/inward/record.url?scp=77956925747&partnerID=8YFLogxK

U2 - 10.1016/S0304-0208(08)73003-2

DO - 10.1016/S0304-0208(08)73003-2

M3 - Article

VL - 115

SP - 141

EP - 149

JO - North-Holland Mathematics Studies

JF - North-Holland Mathematics Studies

SN - 0304-0208

IS - C

ER -