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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
##### Keywords:
minor weight; edge weight; minor vertex; planar map; number of edges
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##### References:
 [1] BORODIN O. V.: On the total coloring of planar graphs. J. Reine Angew. Math. 394 (1989), 180-185. · Zbl 0653.05029 [2] JUCOVIČ E.: Strengthening of a theorem about 3 -polytopes. Geom. Dedicata 13 (1974), 233-237. · Zbl 0297.52006 [3] KOTZIG A.: Contribution to the theory of Eulerian polyhedra. (Slovak), Mat.-Fyz. Čas. (Math. Slovaca) 5 (1955), 101-113. [4] STEINITZ K.: Polyeder und Raumcinteilungen. Encyklop. d. math. Wissensch. 3 (1922), 1-139. [5] Teoria Combinatoria. Proc. Intern. Colloq. Rome 1973, Accademia nacionále dei lincei, Roma, 1976.
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