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Structural properties for certain classes of infinite planar graphs

Abstract

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.

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References

  1. 1.

    D. W. Barnette,Trees in polyhedral graphs, Canad. J. Math. 18 (1966), 731–736.

  2. 2.

    N. Dean, R. Thomas and X. Yu,Spanning paths in infinite planar graphs, J. Graph Th. 23 (1996), 163–174.

  3. 3.

    R. Diestel,Graph Theory (2nd Ed.), Springer-Verlag, New York, 2000.

  4. 4.

    H. O. Jung,On spanning 3-trees in infinite 3-connected planar graphs, Comm. Korean Math. Soc. 11 (1996), 1–21.

  5. 5.

    H. O. Jung, [2, 3]-factors in a 3-connected infinite planar graphs, J. Appl. Math. & Computing 10 (2002), 27–40.

  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.

  7. 7.

    D. P. SandersOn Hamiltonian cycles in certain planar graphs, J. Graph Th. 21 (1996), 43–50.

  8. 8.

    R. Thomas and X. Yu,4-connected projective planar graphs are Hamiltonian, J. Comb. Th. (B) 62 (1995), 114–132.

  9. 9.

    C. ThomassenStraight line representations of infinite planar graphs, J. London Math. Soc. 16 (1977), 411–423.

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Author information

Correspondence to Hwan-Ok Jung.

Additional information

This Work is Supported by Hanshin University Research Grants 2003.

Hwan-Ok Jung received his BS degree from Seoul National University, Korea, and MS and Ph. D. from Universitat Hamburg, Germany, under the direction of Prof. Dr. R. Halin. Since 1989 he has been a full professor of Hanshin University. His research interests: hamiltonian paths, spanning trees and factors in infinite graphs.

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Jung, H. Structural properties for certain classes of infinite planar graphs. JAMC 13, 105 (2003). https://doi.org/10.1007/BF02936078

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AMS Mathematics Subject Classification

  • 05C10
  • 05C75

Key words and phrases

  • semicycle structures
  • planar graphs
  • structural characterizations
  • infinite graphs