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Bounds on the number of cycles of length three in a planar graph

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Abstract

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF 1,F 2, …,F r. Letf 1,f 2, …,f r denote the lengths of the cycles bounding the facesF 1,F 2, …,F r respectively. LetC 3(G) be the number of cycles of length three inG. We give bounds onC 3(G) in terms ofp,f 1,f 2, …,f r. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC 3(G) between the maximum and minimum value are actually achieved.

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References

  1. B. Grünbaum,Convex Polytopes, Wiley-Interscience, New York, 1967.

    MATH  Google Scholar 

  2. B. Grünbaum,Polytopes, graphs, and complexes, Bull. Amer. Math. Soc.76 (1970), 1131–1201.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. L. Hakimi and E. F. Schmeichel,On the number of cycles of length k in a maximal planar graph, J. Graph Theory3 (1979), 69–86.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. L. Hakimi and E. F. Schmeichel,The number of triangles in a triangulation of a set of points in the plane, Elem. Math.35 (1980), 137–142.

    MathSciNet  Google Scholar 

  5. F. Harary,Graph Theory, Addison-Wesley, Reading, Mass., 1969.

    Google Scholar 

  6. E. F. Schmeichel and S. L. Hakimi,On planar graphical degree sequences, SIAM J. Appl. Math.32 (1977), 598–609.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Whitney,Non-separable and planar graphs, Trans. Amer. Math. Soc.34 (1932), 339–362.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Northwestern University, 60201, Evanston, IL, USA

    S. L. Hakimi & E. F. Schmeichel

  2. San José State University, 95192, San José, CA, USA

    S. L. Hakimi & E. F. Schmeichel

Authors
  1. S. L. Hakimi
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  2. E. F. Schmeichel
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Additional information

This research was supported in part by the U.S.A.F. Office of Scientific Research, Systems Command, under Grant AFOSR-76-3017 and the National Science Foundation under Grant ENG79-09724.

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Hakimi, S.L., Schmeichel, E.F. Bounds on the number of cycles of length three in a planar graph. Israel J. Math. 41, 161–180 (1982). https://doi.org/10.1007/BF02760664

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  • DOI: https://doi.org/10.1007/BF02760664

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