Ball packings with high chromatic numbers from strongly regular graphs

Inspired by Bondarenko’s counter-example to Borsuk’s conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension q3 − q2 + q with chromatic number q3 + 1, where q is a prime power. This improves the previous lower bound for the chromatic number of ball packings. 1. The problem and previous works A ball packing in d-dimensional Euclidean space is a collection of balls with disjoint interiors. The tangency graph of a ball packing takes the balls as vertices and the tangent pairs as edges. The chromatic number of a ball packing is defined as the chromatic number of its tangency graph. The Koebe–Andreev–Thurston disk packing theorem says that every planar graph is the tangency graph of a 2-dimensional ball packing. The following question is asked by Bagchi and Datta in [BD13] as a higher dimensional analogue of the four-colour theorem: Problem. What is the maximum chromatic number χ(d) over all the ball packings in dimension d? The authors gave d + 2 ≤ χ(d) as a lower bound since it is easy to construct d + 2 mutually tangent balls. By ordering the balls by size, the authors also argued that κ(d) + 1 is an upper bound, where κ(d) is the kissing number for dimension d. However, the case of d = 3 has already been investigated by Maehara [Mae07], who proved that 6 ≤ χ(3) ≤ 13. His construction for the lower bound uses a variation of Moser’s spindle, which is the tangency graph of an unit disk packing in dimension 2 with chromatic number 4, and the following lemma: Lemma. If there is a unit ball packing in dimension d with chromatic number χ, then there is a ball packing in dimension d+ 1 with chromatic number χ+ 2. The technique of Maehara [Mae07] can be easily generalized to higher dimensions and gives d+ 3 ≤ χ(d). Another progress is made by Cantwell in an answer on MathOverflow [Can], who proved that the graph of the halved 5-cube (also called the Clebsch graph) is the tangency graph of a 5dimensional unit ball packing with chromatic number 8. Then the Lemma implies that 10 ≤ χ(6). This argument can be generalized to higher dimensions using a result of Linial, Meshulam and Tarsi [LMT88, Theorem 4.1], and gave d+ 4 ≤ χ(d) for d = 2 − 2. As we have seen, both constructions study the chromatic number of unit ball packings and invoke the Lemma. We will do the same. A unit ball packing can be regarded as a set of points such that the minimum distance between pairs of points is at least 1, then the tangency graph of the packing is the unit-distance graph for these points. The finite version of the Borsuk conjecture can be formulated as follows: the chromatic number of the unit-distance graph for a set of points with maximum distance 1 is at most d+ 1. So the chromatic number problem for unit ball packings is the “opposite” of the Borsuk conjecture. By ordering the unit balls by height, we see that the chromatic number of a unit ball packing is at most one plus the one-side kissing number. 2010 Mathematics Subject Classification. 05C15, 05E30, 52C17.