Extending two theorems of A. Kotzig

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 m₄+2m₅ ≥ 24(1 - g) + 3 ∑_k ≥4 (k - 4)p_kz; as a corollary we get m₅ ≥ 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.