Multipartite tournaments with small number of cycles

Abstract

Volkmann, Discrete Math. 45 (2002) 19-53, posed the following question. Let 4 be less than or equal to m be less than or equal to n. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n - m + 1 cycles of length m? We answer this question in the affirmative. We raise the following problem. Given m e {3, 4, ... n}, find a characterization of strong n-partite tournaments having exactly n - m + 1 cycles of length m.