An inequality concerning edges of minor weight in convex 3-polytopes
Igor Fabrici ; Stanislav Jendrol'
Discussiones Mathematicae Graph Theory, Tome 16 (1996), p. 81-87 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let eij be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20e3,3+25e3,4+16e3,5+10e3,6+6[2/3]e3,7+5e3,8+2[1/2]e3,9+2e3,10+16[2/3]e4,4+11e4,5+5e4,6+1[2/3]e4,7+5[1/3]e5,5+2e5,6≥120; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

Publié le : 1996-01-01
EUDML-ID : urn:eudml:doc:270533 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1024,
     author = {Igor Fabrici and Stanislav Jendrol'},
     title = {An inequality concerning edges of minor weight in convex 3-polytopes},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {16},
     year = {1996},
     pages = {81-87},
     zbl = {0868.52004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1024}
}

Igor Fabrici; Stanislav Jendrol'. An inequality concerning edges of minor weight in convex 3-polytopes. Discussiones Mathematicae Graph Theory, Tome 16 (1996) pp. 81-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1024/

[000] [1] O. V. Borodin, Computing light edges in planar graphs, in: R. Bodendiek, R. Henn, eds., Topics in Combinatorics and Graph Theory (Physica-Verlag, Heidelberg, 1990) 137-144. | Zbl 0705.05023

[001] [2] O. V. Borodin, Structural properties and colorings of plane graphs, Ann. Discrete Math. 51 (1992) 31-37, doi: 10.1016/S0167-5060(08)70602-2. | Zbl 0765.05043

[002] [3] O. V. Borodin, Precise lower bound for the number of edges of minor weight in planar maps, Math. Slovaca 42 (1992) 129-142. | Zbl 0767.05039

[003] [4] O. V. Borodin, Structural properties of planar maps with the minimal degree 5, Math. Nachr. 158 (1992) 109-117, doi: 10.1002/mana.19921580108. | Zbl 0776.05035

[004] [5] O. V. Borodin and D. P. Sanders, On light edges and triangles in planar graph of minimum degree five, Math. Nachr. 170 (1994) 19-24, doi: 10.1002/mana.19941700103. | Zbl 0813.05020

[005] [6] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973) 390-408, doi: 10.1007/BF02764716. | Zbl 0265.05103

[006] [7] B. Grünbaum, Polytopal graphs, in: D. R. Fulkerson, ed., Studies in Graph Theory, MAA Studies in Mathematics 12 (1975) 201-224. | Zbl 0323.05104

[007] [8] B. Grünbaum, New views on some old questions of combinatorial geometry, Int. Teorie Combinatorie, Rome, 1973, 1 (1976) 451-468.

[008] [9] B. Grünbaum and G. C. Shephard, Analogues for tiling of Kotzig's theorem on minimal weights of edges, Ann. Discrete Math. 12 (1982) 129-140. | Zbl 0504.05026

[009] [10] J. Ivančo, The weight of a graph, Ann. Discrete Math. 51 (1992) 113-116, doi: 10.1016/S0167-5060(08)70614-9. | Zbl 0773.05066

[010] [11] J. Ivančo and S. Jendrol', On extremal problems concerning weights of edges of graphs, in: Coll. Math. Soc. J. Bolyai, 60. Sets, Graphs and Numbers, Budapest (Hungary) 1991 (North Holland, 1993) 399-410.

[011] [12] E. Jucovič, Strengthening of a theorem about 3-polytopes, Geom. Dedicata 3 (1974) 233-237, doi: 10.1007/BF00183214. | Zbl 0297.52006

[012] [13] E. Jucovič, Convex 3-polytopes (Veda, Bratislava, 1981, Slovak).

[013] [14] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat.-Fyz. as. SAV (Math. Slovaca) 5 (1955) 101-103 (Slovak; Russian summary).

[014] [15] A. Kotzig, From the theory of Euler's polyhedra, Mat. as. (Math. Slovaca) 13 (1963) 20-34 (Russian). | Zbl 0134.19601

[015] [16] O. Ore, The four-color problem (Academic Press, New York, 1967). | Zbl 0149.21101

[016] [17] J. Zaks, Extending Kotzig's theorem, Israel J. Math. 45 (1983) 281-296, doi: 10.1007/BF02804013. | Zbl 0524.05031