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On the minimum size of a point set containing a 5-hole and double disjoint 3-holes

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Abstract

Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set of at least n(k, l,m) points in general position, contains an empty convex k-hole; an empty convex l-hole and an empty convex m-hole, which are all pairwise disjoint. In this paper we prove that n(3, 3, 5) = 12.

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References

  1. P. Erdős, “Some more problems on elementrary geometry,” Austr. Math. Soci. Gaze. 5, 52–54 (1978).

    Google Scholar 

  2. H. Harborth, “Konvexe Fünfeck in ebenen Punktmengen,” Element der Math. 33, 116–118 (1978).

    MATH  MathSciNet  Google Scholar 

  3. J. Horton, “Sets with no empty 7-gons,” Canadian Math. Bull. 26, 482–484 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Gerken, “Empty convex hexagons in planar point sets,” Discrete and Comput. Geom. 39, 239–272 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Nicolás, “The empty hexagon theorem,” Discrete and Comput. Geom. 38, 389–397 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Valtr, “On empty hexagons,” Contemp. Math. 453, 433–441 (2008).

    Article  MathSciNet  Google Scholar 

  7. M. Urabe, “On a partition into convex polygons,” Discrete Appl. Math. 64, 179–191 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Hosono and M. Urabe, “A minimal planar point set with specified disjoint empty convex subsets,” Lecture Notes in Compu. Sci. 4535, 90–100 (2008).

    Article  MathSciNet  Google Scholar 

  9. B. Bhattacharya and S. Das, “On the minimum size of a point set containing a 5-hole and a disjoint 4-hole,” Studia ScientiarumMath. Hunga. 48(4), 445–457 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  10. K. Hosono and M. Urabe, “On the minimal size of a point set containing two non-intersecting empty convex polygons,” Lecture Notes in Compu. Sci. 3742, 117–122 (2005).

    Article  MathSciNet  Google Scholar 

  11. K. Hosono and M. Urabe, “On the number of disjoint convex quadrilaterals for a planar point set,” Computational Geom., Theory and Appl. 20, 97–104 (2001).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Science, Hebei University of Science and Technology, Shijiazhuang, China

    X. S. You & X. L. Wei

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  1. X. S. You
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  2. X. L. Wei
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Correspondence to X. S. You.

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The paper was submitted by the authors for the English version of the journal.

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You, X.S., Wei, X.L. On the minimum size of a point set containing a 5-hole and double disjoint 3-holes. Math Notes 97, 951–960 (2015). https://doi.org/10.1134/S0001434615050314

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  • DOI: https://doi.org/10.1134/S0001434615050314

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