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Componentwise Complementary Cycles in Multipartite Tournaments
Zhi-hong HE, Guo-jun LI, Xue-qin ZHOU
Acta Mathematicae Applicatae Sinica(English Series)
2012, 28 (1):
201-208.
DOI: 10.1007/s10255-012-0135-9
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 D\V (C), where α ≥ 2, let Dc = {Di|Di contains cycles, i = 1, 2, … , α}, Dc = {D1,D2, … ,Dα} \ Dc. If Dc ≠ Ø, then D contains a pair of componentwise complementary cycles.
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