Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 217-227 (1999)

What is the Minimum Length of a
Non-Extendable Lace?

Arnfried Kemnitz, Valeriu Soltan

Abt. Diskrete Mathematik, Techn. Universität Braunschweig, D-38106 Braunschweig, Germany

Mathematical Institute, Academy of Sciences of Moldova, MD-2028 Chisinau, Moldova

Abstract: A family $\{C_1,...,C_n\}$ of pairwise distinct, non-overlapping, congruent circles in the plane form a lace provided $C_i$ touches $C_{i+1}$ for all $i = 1,\ldots,n-1$. If, additionally, $C_n$ touches $C_1$, the lace is named a loop. A lace (loop) $\{C_1,\ldots, C_n\}$ is called extendable if it is properly contained in another lace (respectively, loop). In the paper various problems and results on minimum lengths of non-extendable laces and loops are discussed.

Keywords: finite packing, circles, plane

Classification (MSC91): 52C15

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