Componentwise complementary cycles in multipartite tournaments

Abstract

The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete n-partite digraphs with n ≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let D be a 2-strong n-partite (n ≥ 6) tournament that is not a tournament. Let C be a 3-cycle of D and D \ V (C) be nonstrong. For the unique acyclic sequence D1,D2, ...,Dα of DV (C), where α ≥ 2, let Dc = {Di\Di contains cycles, i = 1, 2, ..., α}, \(D_{\bar c} = \{ D_1 ,D_2 , \cdots ,D_\alpha \} \backslash D_c\). If Dc ≠ ∅, then D contains a pair of componentwise complementary cycles.