### Abstract

A homeomorphically irreducible spanning tree (HIST) of a graph G is a spanning tree without vertices of degree 2 in G. Malkevitch conjectured that every 4-connected plane graph has a HIST. In order to solve Malkevitch-conjecture, it is natural to show the existence of a HIST in 3-connected, internally 4-connected (or essentially 4-connected) plane graphs. In this paper, we construct 3-connected, internally 4-connected plane graphs without HISTs. Consequently, such a strategy does not work when we solve Malkevitch-conjecture.

Original language | English |
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Pages (from-to) | 157-165 |

Number of pages | 9 |

Journal | Ars Combinatoria |

Volume | 141 |

Publication status | Published - 2018 Oct 1 |

Externally published | Yes |

### Keywords

- Homeomorphically irreducible spanning tree (HIST)
- Plane graph

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Nomura, R., & Tsuchiya, S. (2018). Plane graphs without homeomorphically irreducible spanning trees.

*Ars Combinatoria*,*141*, 157-165.