Title: On a conjecture of L. Fejes Tóth and J. Molnár about circle coverings of the plane
Language: English
Authors: Dorninger, Dietmar
Category: Research Article
Issue Date: 2018
Journal: Periodica Mathematica Hungarica
ISSN: 1588-2829
Tóth and Molnár (Math Nachr 18:235–243, 1958) formulated the conjecture that for a given homogeneity q the thinnest covering of the Euclidean plane by arbitrary circles is greater or equal a function S(q). Florian (Rend Semin Mat Univ Padova 31:77–86, 1961) proved that if the covering consists of only two kinds of circles then the conjecture is true supposed that S(q)≤S(1/q) what can be easily verified by a computer. In this paper we consider the general case of circles with arbitrary radii from an interval of the reals. We set up two further functions M0(q) and M1(q) and prove that the conjecture is true if S(q) is less than or equal to S(1 / q), M0(q) and M1(q). As in the case of two kinds of circles this can be readily confirmed by computer calculations. (For q≥0.6 we even do not need the function M1(q) for computer aided comparisons.) Moreover, we obtain Florian’s result in a shorter different way.
Keywords: Covering of the plane; Incongruent discs; Minimum density; Conjecture of Tóth and Molnár; Minimal density of three circles covering a triangle
DOI: 10.1007/s10998-018-0254-z
Library ID: AC15503177
URN: urn:nbn:at:at-ubtuw:3-6620
Organisation: E104 - Institut für Diskrete Mathematik und Geometrie 
Publication Type: Article
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