摘要
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,..., Da of D ／ V（C）, where a 〉 2, let Dc = {Di｜Di contains cycles, i = 1,2,...,a}, Dc = {D1,D2,...,Da} ／ De. If Dc≠ 0, then D contains a pair of componentwise complementary cycles.
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,..., Da of D ／ V（C）, where a 〉 2, let Dc = {Di｜Di contains cycles, i = 1,2,...,a}, Dc = {D1,D2,...,Da} ／ De. If Dc≠ 0, then D contains a pair of componentwise complementary cycles.
基金
Supported by the National Natural Science Foundation of China (No. 10801114)
the Nature Science Foundation of Shandong Province, China (No. ZR2011AL019
No. ZR2011AM005)
