Abstract
The chromatic number of a family of solid balls is the minimum number of colors necessary to color the balls so that mutually tangent balls have different colors. A cycle of solid balls is a cyclic sequence of balls in which each consecutive pair of balls are tangent. A cycle of balls is called knotted if the closed polygonal curve obtained by connecting the centers of consecutive balls with line segments is knotted. (a) How large the chromatic number of a family of balls can be? (b) How many balls are necessary to make a knotted cycle? These problems are considered for (1) a general family of balls, (2) balls all sitting on a table, (3) a general family of unit balls, (4) unit balls lying between a pair of parallel planes at distance ρ + 2 apart. This paper gives a survey of the current state of arts including some original new results.
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Maehara, H. On Configurations of Solid Balls in 3-Space: Chromatic Numbers and Knotted Cycles. Graphs and Combinatorics 23 (Suppl 1), 307–320 (2007). https://doi.org/10.1007/s00373-007-0702-7
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DOI: https://doi.org/10.1007/s00373-007-0702-7