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Precise lower bound for the number of edges of minor weight in planar maps. (English) Zbl 0767.05039
Given a planar map, let \(e_{i,k}\) be the number of edges joining a vertex of degree \(i\) with a vertex of degree \(k\). Then, for each simplicial 3-polytope it is shown that \(20e_{3,3}+25e_{3,4}+16e_{3,5}+10e_{3,6}+6{2\over 3} e_{3,7}+5e_{3,8}+2{1\over 2} e_{3,9}+2e_{3,10}+16{2\over 3} e_{4,4}+11e_{4,5}+5e_{4,6}+1{1\over 2} e_{4,7}+5{1\over 3} e_{5,5}+2e_{5,6}\geq 120\). A list of examples shows that these coefficients are the best possible. Similar inequalities are obtained for some particular classes of normal planar maps. These results clarify a problem raised by E. Jucovič in 1974.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
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References:
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