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Indecomposable Coverings with Homothetic Polygons

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Abstract

We prove that for any convex polygon \(S\) with at least four sides, or a concave one with no parallel sides, and any \(m>0\), there is an \(m\)-fold covering of the plane with homothetic copies of \(S\) that cannot be decomposed into two coverings.

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References

  1. 1.

    Keszegh, B., Pálvölgyi, D.: Octants are cover-decomposable. Discrete Comput. Geom. 47(3), 598–609 (2012)

  2. 2.

    Keszegh, B., Pálvölgyi, D.: Convex polygons are self-coverable. Discrete Comput. Geom. 51(4), 885–895 (2014)

  3. 3.

    Pach, J.: Kolloquium über Diskrete Geometrie. Decomposition of Multiple Packing and Covering. University of Salzburg, Salzburg (1980)

  4. 4.

    Pach, J.: Covering the plane with convex polygons. Discrete Comput. Geom. 1(1), 73–81 (1986)

  5. 5.

    Pach, J., Pálvölgyi, D., Tóth, G.: Survey on the decomposition of multiple coverings. In: I. Bárány, et al. (eds.) Geometry-Intuitive, Discrete and Convex. Bolyai Society Mathematical Studies, vol. 24, pp. 219–259. Springer, Heidelberg (2013)

  6. 6.

    Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Discrete Geometry, Combinatorics and Graph Theory, Springer Lecture Notes in Computer Science, vol. 4381, pp. 135–148. Springer, Berlin (2007)

  7. 7.

    Pálvölgyi, D.: Indecomposable coverings with concave polygons. Discrete Comput. Geom. 44(3), 577–588 (2010)

  8. 8.

    Pálvölgyi, D.: Indecomposable coverings with unit discs. arXiv preprint arXiv:1310.6900 (2013)

  9. 9.

    Pálvölgyi, D., Tóth, G.: Convex polygons are cover-decomposable. Discrete Comput. Geom. 43(3), 483–496 (2010)

  10. 10.

    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Minkowski Addition, vol. 44, 3rd edn. Cambridge University Press, Cambridge (1993)

  11. 11.

    Tardos, G., Tóth, G.: Multiple coverings of the plane with triangles. Discrete Comput. Geom. 38(2), 443–450 (2007)

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Acknowledgments

I would like to thank my supervisor Géza Tóth for all the help and for the many discussions. Without him this article would not have been completed. This work has been supported by OTKA NN-102029.

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Correspondence to István Kovács.

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Editor in Charge: János Pach

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Kovács, I. Indecomposable Coverings with Homothetic Polygons. Discrete Comput Geom 53, 817–824 (2015). https://doi.org/10.1007/s00454-015-9687-9

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Keywords

  • Homothetic copy
  • Multiple covering
  • Decomposable