Strongly quasi-Hamiltonian-connected multipartite tournament

1. Strongly quasi-Hamiltonian-connected multipartitetournament 陆玫 清华大学数学科学系 Work joined with Guo Yubao， Lehrstuhl C für Mathematik, RWTH Aachen University 第五届海峡两岸图论与组合学学术会议

2. Terminology and notation D=(V, A):a finite digraph D with the vertex-set V (D) and the arcset A(D) without multiple arcs and loops. orx dominates y or y is dominated by x. 第五届海峡两岸图论与组合学学术会议

3. Considered classes of digraphs • Semicomplete digraphs: • A digraph D is semicomplete if for any two differentvertices x and y of D there is at least one arc betweenthem. 第五届海峡两岸图论与组合学学术会议

4. Tournaments: • A digraph D is called tournament if for any two different vertices x and y of D，there is exactly one arcbetween them. • or a tournament is an orientation of a complete graph. • or a tournament is a semicomplete digraph withoutcycles of length 2. 第五届海峡两岸图论与组合学学术会议

5. Semicomplete n-partite digraphs: • A semicomplete n-partite digraph D consists of n disjoint vertex sets V1, V2, …,Vnsuch that for every pair x, y of vertices, the following conditions are satisfied: • (1) x and y are non-adjacent, if x, y ∈ Vi, 1 ≤ i ≤ n; • (2) there is at least one arc between x and y, if x ∈Vi and y∈Vjwith i ≠j, 1≤i, j ≤n. 第五届海峡两岸图论与组合学学术会议

6. multipartite or n-partite tournament : A multipartite or n-partite tournament is an orientationof a complete c-partite graph or a semicomplete multipartite digraph without a cycle of length 2. 第五届海峡两岸图论与组合学学术会议

7. Locally semicomplete digraph: A digraph D is locally semicomplete if andare both semicomplete for every vertex Local tournament: A locally semicomplete digraph without a 2-cycle iscalled local tournament. 第五届海峡两岸图论与组合学学术会议

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9. A digraph D is strong if for any two vertices x, y of D ,there are a (directed) path from x to y and a (directed)path from y to x. • paths in digraphs : directed papths • cycles in digraphs : directed cycles • a l-cycle : a cycle of length l • D is called k-connected if |V (D)|≥ k +1 and the deletion of any set of fewer than k vertices leaves a strongsubdigraph. 第五届海峡两岸图论与组合学学术会议

10. A path containing all vertices of a digraph D is called a hamiltonian path of D. • A cycle containing all vertices of a digraph D is called a hamiltonian cycle of D. • A digraph D with n ≥3 vertices is called pancyclic if D has a l-cycle for all l satisfying 3 ≤ l ≤ n. 第五届海峡两岸图论与组合学学术会议

11. Tournament • Theorem 2.1 (Rédei, 1934). Every tournament contains a hamiltonian path. • Theorem 2.2 (Camion, 1959). Every strong tournament contains a hamiltonian cycle. • Theorem 2.3 (Harary & Moser, 1966). Every strong tournament T with n vertices is pancyclic. 第五届海峡两岸图论与组合学学术会议

12. A digraph is strongly hamiltonian-connected, if for any two vertices x and y of D, there is a hamiltonian path from x to y and from y to x. • Theorem 2.4 (Thomassen, 1980). Every 4-connected tournament is strongly hamiltonian-connected. 第五届海峡两岸图论与组合学学术会议

13. Locally semicomplete digraphs • Theorem 3.1. (Bang-Jensen, 1991) A connected locally semicomplete digraph has a hamiltonian path. • A strong locally semicomplete digraph has a hamiltonian cycle. 第五届海峡两岸图论与组合学学术会议

14. Theorem 3.2 (C.-Q. Zhang and C. Zhao, 1995). If a locally semicomplete digraph D on n vertices contains a locally strongly connected vertex v, then D is pancyclic and v is contained in cycles of all lengths 3,4,…, n. A vertex v of a digraph D is locally strongly connected if is strong. 第五届海峡两岸图论与组合学学术会议

15. Theorem 3.3 (Guo, 1995). Every 4-connected locally semicomplete digraph is strongly hamiltonian-connected. 第五届海峡两岸图论与组合学学术会议

16. Theorem 4.1 (Bondy, 1976).(1) Every strong semicomplete n-partite (n ≥3) digraph contains a k-cycle for all . (2) If D is a strong semicomplete n-partite (n≥5) digraph, in which each partite set has at least two vertices, then D contains a k-cycle for some k > n. Multipartite tournament 第五届海峡两岸图论与组合学学术会议

17. Problem (Bondy, 1976). Let D be a strong n-partite (n≥5) tournament, in which each partite set has at least 2 vertices. Does D contains an (n + 1)-cycle? • Theorem 4.2 (Guo & Volkmann, 1996). Let D be a strong n-partite (n ≥5) tournament, each of whose partite sets has at least 2 vertices. Then D has no (n+1)-cycle if and only if D is isomorphic to a member of Wm, where m −1 is the diameter of D. 第五届海峡两岸图论与组合学学术会议

18. Theorem 4.3 (Yeo, 1997). Every regular multipartite tournament is hamiltonian. • Theorem 4.4 (Goddard & Oellermann, 1991). Every vertex of a strong semicomplete n-partite (n≥3) digraph is in a cycle that contains vertices from exactly m partite sets for all m with 3≤m ≤ n. 第五届海峡两岸图论与组合学学术会议

19. Theorem 4.5 (Guo & Volkmann, 1994). Let D be a strongly connected n-partite (n≥3) tournament. Then every partite set of D has at least one vertex which lies on an m-cycle for all . Theorem 4.6 (Guo & Volkmann, 1998). Let D be a strongly connected n-partite (n≥3) tournament. Then every partite set of D has at least one vertex which lies on an m-cycle Cm for all such that 第五届海峡两岸图论与组合学学术会议

20. Let D be a n-partite tournament. D is called strongly quasi-Hamiltonian-connected, if for any two vertices x and y of D, there is a path with at least one vertex from each partite set from x to y and from y to x. 第五届海峡两岸图论与组合学学术会议

21. Strongly quasi-Hamiltonian-connected multipartite tournament Lemma 1(Tewes and Volkmann, 1999) Let D be a connected, non-strong c-partite tournament with partite sets V1, V2, … , Vc. Then there exists a unique decomposition of V (D) into pairwise disjoint subsets X1, X2, … , Xr, where Xi is the vertex set of a strong component of D or for some such that for 1≤ i < j≤ r and there are and such that xi→xi+1for 1≤i < r. We use to denote that there is no arc from Y to X. 第五届海峡两岸图论与组合学学术会议

22. Lemma 2 (Guo and Lu, 2009) Let D be a connected, non-strong c-partite tournament with partite sets V1, V2, … , Vc. Let X1, X2, … , Xr be the unique decomposition of V (D) defned as Lemma 1. Then for any and any , D has a path with at least one vertex from each partite set from x1to xr. 第五届海峡两岸图论与组合学学术会议

23. Lemma 3 (Guo and Lu) Let D be a c-partite tournament and D’ be a maximal spanning acyclic subdigraph of D. Then D’ has a path with at least one vertex from each partite set. A digraph is acyclic if it contains no cycle. A spanning subdigraph D’ of a digraph D is maximal if D contains no spanning subdigraph D” with and |E(D’)|< |E(D”)|. 第五届海峡两岸图论与组合学学术会议

24. Theorem 4 (Guo and Lu, 2009) Let D be a c-partite tournament and x, y two distinct vertices of D. If D has a spanning acyclic subdigraph D’ such that for each vertex z of D, D’ contains a path from x to z and a path from z to y, then D has a path from x to y with at least one vertex from each partite set. 第五届海峡两岸图论与组合学学术会议

25. Theorem 5 (Guo and Lu, 2009) Let D be a 2-connected c-partite tournament with partite sets V1, V2, … , Vc and let x, y be two distinct vertices of D. If D contains three internally disjoint (x; y)-paths, each of which is at least 2, then D contains a path from x to y with at least one vertex from each partite set. 第五届海峡两岸图论与组合学学术会议

26. Corollary 6 A 4-connected c-partite tournament is strongly quasi-Hamiltonian-connected. • Corollary 7 (Thomassen, 1980) Every 4-connected tournament is strongly Hamiltonian-connected. 第五届海峡两岸图论与组合学学术会议

27. Problem 1 • Let D be a c-partite tournament. D is called weakly quasi-Hamiltonian-connected, if for any two vertices x and y of D, there is a path with at least one vertex from each partite set from x to y or from y to x. • Problem 1 In what condition, D is weakly quasi-Hamiltonian-connected. 第五届海峡两岸图论与组合学学术会议

28. Problem 2 • Let D be a c-partite tournament. D is called strongly pseudo-Hamiltonian-connected, if for any two vertices x and y of D, there is a path of length c+1 from x to y and from y to x. • Conjecture:A 4-connected c-partite tournament is strongly pseudo-Hamiltonian-connected. 第五届海峡两岸图论与组合学学术会议

29. Thank you for your attention! 第五届海峡两岸图论与组合学学术会议