第五届海峡两岸图论与组合学学术 会议 1 Strongly quasi-Hamiltonian- connected multipartite tournament 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 arc set A(D) without multiple arcs and loops. or x 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 different vertices x and y of D there is at least one arc between them.

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 arc between them.  or a tournament is an orientation of a complete graph.  or a tournament is a semicomplete digraph without cycles 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 V 1, V 2, …,V n such that for every pair x, y of vertices, the following conditions are satisfied:  (1) x and y are non-adjacent, if x, y ∈ V i, 1 ≤ i ≤ n;  (2) there is at least one arc between x and y, if x ∈ V i and y ∈ V j with 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 orientation of 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 and are both semicomplete for every vertex Local tournament:  A locally semicomplete digraph without a 2-cycle is called local tournament.

第五届海峡两岸图论与组合学学术 会议 8

第五届海峡两岸图论与组合学学术 会议 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 strong subdigraph.

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? m  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 W m, 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 C m 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 V 1, V 2, …, V c. Then there exists a unique decomposition of V (D) into pairwise disjoint subsets X 1, X 2, …, X r, where X i 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 x i →x i+1 for 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 V 1, V 2, …, V c. Let X 1, X 2, …, X r 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 x 1 to x r.

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 V 1, V 2, …, V c 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!