### Abstract

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | Universiteit Twente |

Number of pages | 7 |

ISBN (Print) | 0169-2690 |

Publication status | Published - 1998 |

### Publication series

Name | Memorandum / University of Twente, Faculty of Applied Mathematics, ISSN 0921-1969 ; no. 1427 |
---|---|

Publisher | Universiteit Twente |

### Keywords

- MSC-05C35
- MSC-05C38
- IR-30611
- EWI-3247
- METIS-141252
- MSC-05C45

### Cite this

*A note on a conjecture concerning tree-partitioning 3-regular graphs*. (Memorandum / University of Twente, Faculty of Applied Mathematics, ISSN 0921-1969 ; no. 1427). Enschede: Universiteit Twente.

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*A note on a conjecture concerning tree-partitioning 3-regular graphs*. Memorandum / University of Twente, Faculty of Applied Mathematics, ISSN 0921-1969 ; no. 1427, Universiteit Twente, Enschede.

**A note on a conjecture concerning tree-partitioning 3-regular graphs.** / Bohme, T.; Bohme, T.; Broersma, Haitze J.; Tuinstra, Hilde.

Research output: Book/Report › Report › Professional

TY - BOOK

T1 - A note on a conjecture concerning tree-partitioning 3-regular graphs

AU - Bohme, T.

AU - Bohme, T.

AU - Broersma, Haitze J.

AU - Tuinstra, Hilde

N1 - Imported from MEMORANDA

PY - 1998

Y1 - 1998

N2 - If $G$ is a 4-connected maximal planar graph, then $G$ is Hamiltonian (by a theorem of Whitney), implying that its dual graph $G^*$ is a cyclically 4-edge connected 3-regular planar graph admitting a partition of the vertex set into two parts, each inducing a tree in $G^*$, a so-called tree-partition. It is a natural question whether each cyclically 4-edge connected 3-regular graph admits such a tree-partition. This was conjectured by Jaeger, and recently independently by the first author. The main result of this note shows that each connected 3-regular graph on $n$ vertices admits a partition of the vertex set into two sets such that precisely $n/2+2$ edges have end vertices in each set. This is a necessary condition for having a tree-partition. We also show that not all cyclically 3-edge connected 3-regular (planar) graphs admit a tree-partition, and present the smallest counterexamples.

AB - If $G$ is a 4-connected maximal planar graph, then $G$ is Hamiltonian (by a theorem of Whitney), implying that its dual graph $G^*$ is a cyclically 4-edge connected 3-regular planar graph admitting a partition of the vertex set into two parts, each inducing a tree in $G^*$, a so-called tree-partition. It is a natural question whether each cyclically 4-edge connected 3-regular graph admits such a tree-partition. This was conjectured by Jaeger, and recently independently by the first author. The main result of this note shows that each connected 3-regular graph on $n$ vertices admits a partition of the vertex set into two sets such that precisely $n/2+2$ edges have end vertices in each set. This is a necessary condition for having a tree-partition. We also show that not all cyclically 3-edge connected 3-regular (planar) graphs admit a tree-partition, and present the smallest counterexamples.

KW - MSC-05C35

KW - MSC-05C38

KW - IR-30611

KW - EWI-3247

KW - METIS-141252

KW - MSC-05C45

M3 - Report

SN - 0169-2690

T3 - Memorandum / University of Twente, Faculty of Applied Mathematics, ISSN 0921-1969 ; no. 1427

BT - A note on a conjecture concerning tree-partitioning 3-regular graphs

PB - Universiteit Twente

CY - Enschede

ER -