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1996 | 16 | 2 | 123-141
Tytuł artykułu

Unavoidable set of face types for planar maps

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The type of a face f of a planar map is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f. A set 𝒯 of face types is found such that in any normal planar map there is a face with type from 𝒯. The set 𝒯 has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps.
Wydawca
Rocznik
Tom
16
Numer
2
Strony
123-141
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-04-09
Twórcy
  • Department of Geometry and Algebra, P. J. Šafárik University, Jesenná 5, 041 54 Košice, Slovakia
  • Department of Geometry and Algebra, P. J. Šafárik University, Jesenná 5, 041 54 Košice, Slovakia
Bibliografia
  • [1] O.V. Borodin, On the total coloring of planar graphs, J. reine angew. Math. 394 (1989) 180-185, doi: 10.1515/crll.1989.394.180.
  • [2] O.V. Borodin, Precise lower bound for the number of edges of minor weight in planar maps, Math. Slovaca 42 (1992) 129-142.
  • [3] O.V. Borodin, A structural theorem on planar maps and its application to coloring, Diskret. Mat. 4 (1992) 60-65. (Russian)
  • [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.
  • [5] O.V. Borodin, Joint extension of two theorems of Kotzig on 3-polytopes, Combinatorica 13 (1993) 121-125, doi: 10.1007/BF01202794.
  • [6] O.V. Borodin, The structure of edge neighbourhoods in plane graphs and the simultaneous coloring of vertices, edges and faces, Mat. Zametki 53 (1993) 35-47. (Russian)
  • [7] O.V. Borodin, Simultaneous coloring of edges and faces of plane graphs, Discrete Math. 128 (1994) 21-33, doi: 10.1016/0012-365X(94)90101-5.
  • [8] O.V. Borodin, Triangles with restricted degree sum of their boundary vertices in plane graphs, Discrete Math. 137 (1995) 45-51, doi: 10.1016/0012-365X(94)E0144-7.
  • [9] O.V. Borodin and D. P. Sanders, On light edges and triangles in planar graphs of minimum degree five, Math. Nachr. 170 (1994) 19-24, doi: 10.1002/mana.19941700103.
  • [10] B. Grünbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973) 390-408, doi: 10.1007/BF02764716.
  • [11] B. Grünbaum, New views on some old questions of combinatorial geometry, in.: Proc. Internat. Colloq. Combin. Theory (Rome, 1973) Accad. Naz. Lincei Rome (1976) 451-468.
  • [12] B. Grünbaum and G.C. Shephard, Analogues for tilings of Kotzig's theorem on minimal weights of edges, Ann. Discrete Math. 12 (1982) 129-140.
  • [13] J. Ivančo, The weight of a graph, Ann. Discrete Math. 51 (1992) 113-116, doi: 10.1016/S0167-5060(08)70614-9.
  • [14] S. Jendrol' and Z. Skupień, On the vertex/edge distance colourings of planar graphs, submitted.
  • [15] T.R. Jensen and B. Toft, Graph Coloring Problems (John Wiley&Sons, Inc. New York, 1995).
  • [16] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat.-Fyz. Casopis Sloven. Akad. Vied 5 (1955) 101-103. (Slovak)
  • [17] A. Kotzig, On the theory of Euler polyhedra, Mat.-Fyz. Casopis Sloven. Akad. Vied 13 (1963) 20-31. (Russian)
  • [18] A. Kotzig, Extremal polyhedral graphs, Ann. New York Acad. Sci. 319 (1979) 569-570.
  • [19] H. Lebesgue, Quelques conséquences simples de la formule d'Euler, J. Math. Pures Appl. 19 (1940) 19-43.
  • [20] O. Ore and M.D. Plummer, Cyclic coloration of plane graphs, in: Recent Progress in Combinatorics (Proceedings of the Third Waterloo Conference on Combinatorics), Academic Press, New York (1969) 287-293.
  • [21] M.D. Plummer and B. Toft, Cyclic coloration of 3-polytopes, J. Graph Theory 11 (1987) 507-515, doi: 10.1002/jgt.3190110407.
  • [22] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968.
  • [23] J. Zaks, Extending Kotzig's theorem, Israel J. Math. 45 (1983) 281-296, doi: 10.1007/BF02804013.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1028
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