A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in R3 is not greater than 13.955. We also find new upper bounds for the average degree of contact graphs in R4 and R5.
Glazyrin, A. (2020). Contact graphs of ball packings. Journal of Combinatorial Theory, Series B, 145, 323–340. https://doi.org/10.1016/j.jctb.2020.05.007
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Journal of Combinatorial Theory, Series B