## Abstract

Two theorems of A. Kotzig are extended, as follows: 1. (1) A. Kotzig proved in 1963 that every 5-valent 3-connected planar graph contains a vertex which meets at least four triangles. We prove that if a 5-valent 3-connected graph on the orientable surface of genus g has p_{k} k-gons, k ≥3, and m_{i} vertices meeting precisely i triangles, 0 ≤ i ≤ 5, then m_{4}+2m_{5} ≥ 24(1 - g) + 3 ∑_{k} ≥4 (k - 4)p_{k}z; as a corollary we get m_{5} ≥ 12(1 - g) + ∑_{k}≥4 (k - 6)p_{k}. 2. (2) Let σ(e) denote the sum of the valences of the two faces incident with the edge e of a planar graph. A Kotzig proved that every 4-valent 3-connected planar graph has an edge e such that δ(e) ≤ 8. We use Kotzig's proof to get the bound 9 in the corresponding toroidal case.

Original language | English |
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Pages (from-to) | 309-315 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 43 |

Issue number | 2-3 |

DOIs | |

State | Published - 1983 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics