2009-7-29 第五届海峡两岸图论与组合学学术会议 1 strongly quasi-hamiltonian-...

2009-7-29 第五届海峡两岸图论与组合学学术会议

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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


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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.

yxDAxy :)(or x dominates y or y is dominated by x.

)(|)(

)(|)(

DAyxyxN

DAxyyxN


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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.


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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.


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Semicomplete n-partite digraphs:

A semicomplete n-partite digraph D consists of n disjoint vertex sets V1, V2, …,Vn such 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∈Vj with i ≠j, 1≤i, j ≤ n.


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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.


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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.

)(xND )(xND)(DVx


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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.


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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.


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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.


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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.


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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.


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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.

vvNvND )()(


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Theorem 3.3 (Guo, 1995). Every 4-connected locally semicomplete digraph is strongly hamiltonian-connected.


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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

nk ,,4,3


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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 Wmm, where m − 1 is the diameter of D.


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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.


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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

nm ,,4,3

nm ,,4,3

nCVCVCV 43


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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.


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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+1 for 1≤i < r.

We use to denote that there is no arc from Y to X.

li VX },,4,3{ cl

ji XX ii Xx 11 ii Xx

YX


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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 x1 to xr.

11 Xx rr Xx


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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”)|.DD


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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.


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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.


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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.


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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.


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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.


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Thank you for your attention!

2009-7-29 第五届海峡两岸图论与组合学学术 会议 1 strongly quasi-hamiltonian-...

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2009-7-29 第五届海峡两岸图论与组合学学术会议 1 strongly quasi-hamiltonian-...