Geodesic
Enumeration of labelled and unlabelled Hamiltonian cycles in complete $k$-partite graphs
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 119-142

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We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Turán graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$. 
@article{ZNSL_2019_488_a5,
     author = {E. C. Krasko and I. N. Labutin and A. V. Omelchenko},
     title = {Enumeration of labelled and unlabelled {Hamiltonian} cycles in complete $k$-partite graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--142},
     publisher = {mathdoc},
     volume = {488},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a5/}
}
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E. C. Krasko; I. N. Labutin; A. V. Omelchenko. Enumeration of labelled and unlabelled Hamiltonian cycles in complete $k$-partite graphs. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 119-142. http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a5/