Almost Symmetric Cycles in Large Digraphs
- 40 Downloads
We prove that every digraph D with n≥7, n≥ Open image in new window +6 vertices and at least (n−k−1)(n−1)+k(k+1) arcs contains all symmetric cycles of length at most n−k−2, an almost symmetric cycle of length n−k−1, and with some exceptions, also an almost symmetric cycle of length n−k. Consequently, D contains all orientations of cycles of length at most n−k, unless D is an exception.
KeywordsGraph Digraph Cycle Orientation of cycle Symmetric cycle
AMS Subject Classification05C20
Unable to display preview. Download preview PDF.
- 1.Bermond, J. C., Thomassen, C.: Cycles in digraphs - a survey, J. Graph Theory 5, 1–43 (1981)Google Scholar
- 2.Czaplicki, D. G.: Cykle w turniejach. Hipoteza Rosenfelda dla cykli. (Cycles in tournaments. The conjecture of Rosenfeld for cycles. - in Polish), Master Thesis, Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, 2003Google Scholar
- 3.Erdös, P.: Remarks on a paper of Pósa, Magyar Tud. Akad. Kutatö Int. Közl. 7, 227–229 (1962)Google Scholar
- 4.Havet, F.: Oriented hamiltonian cycles in tournaments, J. Combin. Theory Ser. B 80, 1–31 (2000)Google Scholar
- 5.Heydemann, M. C., Sotteau, D.: Number of arcs and cycles in digraphs, Discrete Math. 52, 199–207 (1984).Google Scholar
- 6.Heydemann, M. C., Sotteau, D., Thomassen, C.: Orientations of hamiltonian cycles in digraphs, Ars. Combin. 14, 3–8 (1982)Google Scholar
- 7.Rosenfeld, M.: Antidirected hamiltonian circuits in tournaments, J. Combin. Theory Ser. B 16, 234–242 (1974)Google Scholar
- 8.Thomason, A.: Paths and cycles in tournaments, Trans. Amer. Math. Soc. 296, 167–180 (1986)Google Scholar
- 9.Wojda, A. P.: Orientations of hamiltonian cycles in large digraphs, J. Graph Theory 10, 211–218 (1986)Google Scholar
- 10.Woodall, D. R.: Sufficient conditions for circuits in graphs, Proc. London Math. Soc. 24, 739–755 (1972)Google Scholar