Geodesic
On the n-Partite Tournaments with Exactly n − m + 1 Cycles of Length m
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 75-82

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Gutin and Rafiey [Multipartite tournaments with small number of cycles, Australas J. Combin. 34 (2006) 17–21] raised the following two problems: (1) Let m ∈ 3, 4, . . ., n. Find a characterization of strong n-partite tournaments having exactly n − m + 1 cycles of length m; (2) Let 3 ≤ m ≤ n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n − m + 1 cycles of length m for two values of m? In this paper, we discuss the strong n-partite tournaments D containing exactly n − m + 1 cycles of length m for 4 ≤ m ≤ n − 1. We describe the substructure of such D satisfying a given condition and we also show that, under this condition, the second problem has a negative answer.
Keywords: multipartite tournaments, tournaments, cycles 
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Guo, Qiaoping; Meng, Wei. On the n-Partite Tournaments with Exactly n − m + 1 Cycles of Length m. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 75-82. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a4/