TITLE: Chromatic number of graphs of tangent closed balls
QUESTION [16 upvotes]: The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a disk packing whose graph is $G$. What happens when disks are replaced by closed balls? By closed balls of higher dimension? I have already asked one question about this here:
Graphs of Tangent Spheres
The question I want to ask here is what is known about the chromatic numbers of these graphs? I have updated the numbers and changed the arguments in the following based on some of the answers.
Assume the chromatic number is 14 or more and we have the smallest such graph that is colorable with 14 or more colors. Take one of the smallest closed balls then since the kissing number for three dimensions is 12 there are at most 12 closed balls tangent to this closed ball. Remove this closed ball then the remaining graph can be colored in 13 or less colors. Color it with 13 colors. Then add the closed ball back in since it is tangent to only 12 closed balls it can be given one of the thirteen colors so we have the entire graph can be colored with thirteen colors which gives a contradiction so the chromatic number must be 13 or less. We have an lower bound of 6 from a spindle constructed according to David Eppstein's answer. Can we improve on the 6 to 13 range? 
We have the lower bound is a quadratic function and we have an upper bound that is exponential. Which of these two is right?  
Is there a case where closed balls of different sizes raise the chromatic number from closed balls the same size?
Finally based on the existing chromatic numbers I am wondering if it is possible to answer this question. Is there a dimension where the chromatic number of the unit distance graph is different from the chromatic number of the graphs in that dimension of tangent closed balls. The unit distance graph is the set of all points in the $n$-dimensional space with two points connected if their distance is one. For dimension two the chromatic number is known to be in the range from 4 to 7. For dimension three the range is 6 to 15. For the graphs of tangent disks we have a chromatic number of 4 and for closed balls a range from 6 to 13. So the possibility that the chromatic numbers of the two types of graphs are the same has not yet been eliminated. So the specific question is what is known and what can be proved about the chromatic number of the graphs of tangent closed balls?

REPLY [12 votes]: Update May 2016
I removed the updates in Oct 2015.  I was trying to combine two copies of strongly regular ball packings to double the chromatic number.  But it has been point out that my construction was buggy.
Previous examples obtained in January 2015
Inspired by Bodarenko's counter-example to the Borsuk conjecture, I recently find many ball packings whose chromatic number is significantly higher than the dimension.  There tangency graphs are all strongly regular.  A note is available on arXiv and here (up to date).  
Here I list their parameters, dimensions and lower bounds (Hoffman) for the chromatic numbers.  Many of these parameters are for the complement to the more famous graph, e.g. Higman-Sims graph.

$(100, 77, 60, 56)$ (Higman-Sims graph), dimension 22, $\chi\ge 80/3$.
$(105, 72, 51, 45)$, dimension 20, $\chi\ge 25$.
$(120, 77, 52, 44)$, dimension 20, $\chi\ge 80/3$.
$(126, 75, 48, 39)$, dimension 20, $\chi\ge 26$.
$(162, 105, 72, 60)$ (Local McLaughlin), dimension 21, $\chi\ge 36$.
$(175, 102, 65, 51)$, dimension 21, $\chi\ge 35$.
$(176, 105, 68, 54)$, dimension 21, $\chi\ge 36$.
$(176, 85, 48, 34)$, dimension 22, $\chi\ge 88/3$.
$(243, 132, 81, 60)$ (Delsarte graph), dimension 22 $\chi\ge 45$.
$(253, 140, 87, 65)$, dimension 22, $\chi\ge 143/3$.
$(275, 162, 105, 81)$ (McLaughlin graph), dimension 22, $\chi\ge 55$.
$(276, 135, 78, 54)$, dimension 23, $\chi\ge 46$.
$(729, 520, 379, 350)$, dimension 112, $\chi\ge 621/5$.

Furthermore, there are two infinite families with high chromatic number (here $q$ is a prime power).

$(q^3, (q+1)(q^2-1)/2, (q+3)(q^2-3)/4+1, (q+1)(q^2-1)/4)$ (complements to Hubaut's C20), dimension $q^2-q$, $\chi\ge q^2$.
$((q^3+1)(q+1), q^4, (q^2+1)(q-1)q, q^3(q-1))$ (complement to the point graph of the generalized quadrangle $(q,q^2)$), dimension $q^3-q^2+q$, $\chi=q^3+1$.

Note that the last case is an equality.  This is the first non-constant lower bound for $\chi-d$.
Hope this helps future improvement.