I would like to know what is curently known about the number $N$ of touching-triplets involving a given vertex $V$ in the contact-graph of $d$-hypersphere packings.
For $d=2$ and $d=3$, one has $N = D(d-1)$ where $D$ is the degree of $V$, if you don't consider the orientation of the $3$-cycles. Note that
- in those cases, it does not require space to be Euclidian.
- this works also for hypercubic lattice graphs if you replace touching triplets with $4$-cycles, which correspond to the shortest non-trivial cycles on the graph in that case.
How does this extend to higher dimensions? In other words, is $1+N/D$ a reliable estimate of the (local) dimension of the lowest-dimension space in which a graph may be embedded ?