Extending two theorems of A. Kotzig

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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 pk k-gons, k ≥3, and mi vertices meeting precisely i triangles, 0 ≤ i ≤ 5, then m4+2m5 ≥ 24(1 - g) + 3 ∑k ≥4 (k - 4)pkz; as a corollary we get m5 ≥ 12(1 - g) + ∑k≥4 (k - 6)pk. 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 languageEnglish
Pages (from-to)309-315
Number of pages7
JournalDiscrete Mathematics
Issue number2-3
StatePublished - 1983

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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