On a conjecture of L. Fejes Tóth and J. Molnár about circle coverings of the plane

Abstract

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.