SB(3,n) has no Hamiltonian cycle for even n, proven by a sign-of-permutation obstruction that extends to SB(m,n) for all odd m congruent to 3 mod 4.
On a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registers
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$\mathit{SB}(3,n)$ has no Hamiltonian cycle when $n$ is even: a sign-of-permutation proof, with extension to all odd $m\equiv 3\pmod 4$
SB(3,n) has no Hamiltonian cycle for even n, proven by a sign-of-permutation obstruction that extends to SB(m,n) for all odd m congruent to 3 mod 4.