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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $d$. 
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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/

[1] B. Nienhuis, “Exact critical point and critical exponents of $O(n)$ models in two dimensions”, Phys. Rev. Lett., 49 (1062), 1062 | DOI | MR

[2] K. A. Dill, “Polymer principles and protein folding”, Protein Sci., 8:6 (1999), 1166–80 | DOI

[3] O. Bodroža-Pantić, B. Pantić, I. Pantić, M. Bodroža-Solarov, “Enumeration of Hamiltonian cycles in some grid graphs”, Commun. Math. Comput. Chem., 70 (2013), 181–204 | MR | Zbl

[4] E. Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, 2014, arXiv: 1402.0545 [math.CO]

[5] J. L. Jacobsen, “Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions”, J. Phys. A: Math. Theor., 40, 14667 | DOI | MR | Zbl

[6] C. Thomassen, “On the number of Hamiltonian cycles in bipartite graphs”, Combinatorics, Probability and Computing, 5 (1996), 437–442 | DOI | MR | Zbl

[7] N. Alon, “The maximum number of Hamiltonian paths in tournaments”, Combinatorica, 10:4 (1990), 319–324 | DOI | MR | Zbl

[8] Endre Szemeredi, G. N. Sarkozya, S. M. Selkowa, “On the number of Hamiltonian cycles in Dirac graphs”, Discrete Mathematics, 265 (2003), 237–250 | DOI | MR | Zbl

[9] A. J. Schwenk, “Enumeration of Hamiltonian cycles in certain generalized Petersen graphs”, J. Combin. Theory Ser. B, 47 (1989), 53–59 | DOI | MR | Zbl

[10] E. Dixon, S. Goodman, “On the number of Hamiltonian circuits in the n-cube”, Proceedings of the American Mathematical Society, 50 (1975), 500–504 | MR | Zbl

[11] D. Singmaster, “Hamiltonian circuits on the n-dimensional octahedron”, J. Combin. Theory, Ser. B, 19:1 (1975), 1–4 | DOI | MR | Zbl

[12] M. Hazewinkel, V. V. Kalashnikov, Counting Interlacing Pairs on the Circle, Report AM. Stichting Mathematisch Centrum, Department of Analysis, Algebra and Geometry, 1995

[13] A. V. Omelchenko, E. S. Krasko, “Enumeration of chord diagrams without loops and parallel chords”, The Electronic Journal of Combinatorics, 24:3 (2017), P3.43 | MR | Zbl

[14] R. J. Mathar, A class of multinomial permutations avoiding object clusters viXra:1511.0015