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Structural properties of planar maps with the minimal degree 5. (English) Zbl 0776.05035
Given a planar map of minimal degree 5, let \(e_{i,j}\) denote the number of edges connecting a vertex of degree \(i\) with a vertex of degree \(j\) and let \(f_{i,j,k}\) denote the number of 3-faces whose vertices have the degrees \(i,j,k\), respectively. Then, if the planar map contains at least one 3-face, the following inequalities hold:
(1) \({18\over 7}e_{5,5}+e_{5,6}\geq 60\),
(2) \(2e_{5,5}+e_{5,6}+{2\over 7}e_{5,7}\geq 60\),
(3) \(18f_{5,5,5}+9f_{5,5,6}+5f_{5,5,7}+4f_{5,6,6}\geq 144\).

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
57M15 Relations of low-dimensional topology with graph theory
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