Hamiltonicity of de Bruijn- and Kautz-based exponential graphs

Determine whether the exponential graphs $K_n^{B(2,k)}$ and $K_n^{K(2,k)}$, formed using the complete graph $K_n$ as base and the de Bruijn network $B(2,k)$ or Kautz network $K(2,k)$ as exponent, are Hamiltonian.

Background

The paper constructs $K_n^{B(2,k)}$ and $K_n^{K(2,k)}$ as maximally connected and super edge-connected graphs of doubly exponential order with logarithmic diameter. Unlike the DCell network, whose Hamiltonicity is known, the Hamiltonicity of these two exponential graph families is explicitly left unresolved in the comparison of their network properties.