Structural properties for certain classes of infinite planar graphs



An infinite locally finite plane graph is called an LV-graph if it is 3-connected and VAP-free. If an LV-graphG contains no unbounded faces, then we say thatG is a 3LV-graph. In this paper, a structure theorem for an LV-graph concerning the existence of a sequence of systems of paths exhausting the whole graph is presented. Combining this theorem with the early result of the author, we obtain a necessary and sufficient conditions for an infinite VAP-free planar graph to be a 3LV-graph as well as an LV-graph. These theorems generalize the characterization theorem of Thomassen for infinite triangulations.

AMS Mathematics Subject Classification

05C10 05C75 

Key words and phrases

semicycle structures planar graphs structural characterizations infinite graphs 


  1. 1.
    D. W. Barnette,Trees in polyhedral graphs, Canad. J. Math. 18 (1966), 731–736.MATHMathSciNetGoogle Scholar
  2. 2.
    N. Dean, R. Thomas and X. Yu,Spanning paths in infinite planar graphs, J. Graph Th. 23 (1996), 163–174.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Diestel,Graph Theory (2nd Ed.), Springer-Verlag, New York, 2000.Google Scholar
  4. 4.
    H. O. Jung,On spanning 3-trees in infinite 3-connected planar graphs, Comm. Korean Math. Soc. 11 (1996), 1–21.MATHGoogle Scholar
  5. 5.
    H. O. Jung, [2, 3]-factors in a 3-connected infinite planar graphs, J. Appl. Math. & Computing 10 (2002), 27–40.MATHCrossRefGoogle Scholar
  6. 6.
    C. St. J. A. Nash-Williams,Unexplored and semi-explored territories in graph theory (ed. F. Harary), Academic press, New York (1973), 149–186.Google Scholar
  7. 7.
    D. P. SandersOn Hamiltonian cycles in certain planar graphs, J. Graph Th. 21 (1996), 43–50.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Thomas and X. Yu,4-connected projective planar graphs are Hamiltonian, J. Comb. Th. (B) 62 (1995), 114–132.CrossRefMathSciNetGoogle Scholar
  9. 9.
    C. ThomassenStraight line representations of infinite planar graphs, J. London Math. Soc. 16 (1977), 411–423.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2003

Authors and Affiliations

  1. 1.Departent of MathematicsHanshin UniversityOsanKorea

Personalised recommendations