  In 1972 Mykkeltveit proved that the maximum number of vertex-disjoint cycles
in the de Bruijn graphs of order $n$ is attained by the pure cycling register
rule, as conjectured by Golomb. We generalize this result to the tensor product
of the de Bruijn graph of order $n$ and a simple cycle of size $k$, when $n$
divides $k$ or vice versa. We also develop counting formulae for a large family
of cycling register rules, including the linear register rules proposed by
Golomb.
