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2004 | 24 | 3 | 403-411
Tytuł artykułu

On the structure of plane graphs of minimum face size 5

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Języki publikacji
EN
Abstrakty
EN
A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is known that a plane graph of minimum face size 5 contains light paths and a light pentagon. In this paper we show that every plane graph of minimum face size 5 contains also a light star $K_{1,3}$ and we present a structural result concerning the existence of a pair of adjacent faces with degree-bounded vertices.
Słowa kluczowe
Wydawca
Rocznik
Tom
24
Numer
3
Strony
403-411
Opis fizyczny
Daty
wydano
2004
otrzymano
2003-01-28
poprawiono
2004-04-16
Twórcy
  • Institute of Mathematics, Faculty of Sciences, University of P.J. Šafárik, Jesenná 5, 041 54 Košice, Slovak Republic
Bibliografia
  • [1] H. Enomoto and K. Ota, Connected Subgraphs with Small Degree Sums in 3-Connected Planar Graphs, J. Graph Theory 30 (1999) 191-203, doi: 10.1002/(SICI)1097-0118(199903)30:3<191::AID-JGT4>3.0.CO;2-X
  • [2] I. Fabrici, On vertex-degree restricted subgraphs in polyhedral graphs, Discrete Math. 256 (2002) 105-114, doi: 10.1016/S0012-365X(01)00368-5.
  • [3] I. Fabrici, J. Harant and S. Jendrol', Paths of low weight in planar graphs, submitted.
  • [4] I. Fabrici, E. Hexel, S. Jendrol' and H. Walther, On vertex-degree restricted paths in polyhedral graphs, Discrete Math. 212 (2000) 61-73, doi: 10.1016/S0012-365X(99)00209-5.
  • [5] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combin. 13 (1997) 245-250.
  • [6] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar graphs, Discrete Math. 191 (1998) 83-90, doi: 10.1016/S0012-365X(98)00095-8.
  • [7] J. Harant, S. Jendrol' and M. Tkáč, On 3-connected plane graphs without triangular faces, J. Combin. Theory (B) 77 (1999) 150-161, doi: 10.1006/jctb.1999.1918.
  • [8] S. Jendrol', T. Madaras, R. Soták and Z. Tuza, On light cycles in plane triangulations, Discrete Math. 197/198 (1999) 453-467.
  • [9] S. Jendrol' and P. Owens, On light graphs in 3-connected plane graphs without triangular or quadrangular faces, Graphs and Combin. 17 (2001) 659-680, doi: 10.1007/s003730170007.
  • [10] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. Cas. SAV (Math. Slovaca) 5 (1955) 111-113.
  • [11] H. Lebesgue, Quelques consequences simples de la formule d'Euler, J. Math. Pures Appl. 19 (1940) 19-43.
  • [12] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1239
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