zbMATH — the first resource for mathematics

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.

05C10 Planar graphs; geometric and topological aspects of graph theory
PDF BibTeX Cite
Full Text: EuDML
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.