Exact diameter of the DCell network

Determine the exact value of the diameter of the DCell network D_{k,n}, beyond the known upper bound ${\rm diam}(D_{k,n})\leq 2^{k+1}-1$.

Background

The DCell network is known to have logarithmic diameter, with the bound diam(Dk,n)≤2k+1−1{\rm diam}(D_{k,n})\leq 2^{k+1}-1. The paper explicitly identifies the exact diameter as unresolved, so the problem is to determine the precise maximum distance between vertices of D_{k,n}.

References

Thus, $D_{k,n}$ has logarithmic diameter; however, the exact value of ${\rm diam}(D_{k,n})$ remains unknown.

— Exponentiation of Graphs  (2501.15716 - Hasunuma, 27 Jan 2025) in Section 1, Introduction