C-partite Tournament Research Articles

A characterization of rich c-partite (c > 7) tournaments without (c + 2)-cycles

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.

Partition of regular balanced c-tournaments into strongly connected c-tournaments

A multipartite tournament will be called r-balanced if every partite set has r vertices. The study of the existence of strongly connected subtournaments in multipartite tournaments started in 1999 by Volkmann. In this paper we study a much more general problem: the existence of a partition of a c-partite tournament into strongly connected tournaments of order c, which will be called strong partition. Given a positive integer r, the strong partition number, ST(r), is the minimum integer c′ such that every regular r-balanced c-partite tournament, with c≥c′, has a strong partition. We prove that for every r≥2, ST(r) exists and that 5≤ST(2)≤7. Moreover, as a consequence of our technique, for every r, there exists a c such that for every c′>c, there exists a strong partition into maximal tournaments of minimum degree ⌊c′−24⌋+1.

On cycles in regular 3-partite tournaments

A c-partite tournament is an orientation of a complete c-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament D is contained in an m-, (m+1)- or (m+2)-cycle for each m∈{3,4,…,|V(D)|−2}, and this conjecture was proved to be correct for 3≤m≤7. In 2012, Xu et al. conjectured that every arc of an r-regular 3-partite tournament D with r≥2 is contained in a (3k−1)- or 3k-cycle for k=2,3,…,r. They proved that this conjecture is true for k=2. In this paper, we confirm this conjecture for k=3, which also implies that Volkmann’s conjecture is correct for m=7,8.

Strong subtournaments and cycles of multipartite tournaments

In this paper we relate the global irregularity and the order of a c-partite tournament T to the existence of certain cycles and the problem of finding the maximum strongly connected subtournament of T. In particular, we give results related to the following problem of Volkmann: How close to regular must a c-partite tournament be, to secure a strongly connected subtournament of order c?

Weakly quasi-Hamiltonian-connected multipartite tournaments

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. In 2013, Lu, Guo and Surmacs introduced the concept of quasi-Hamiltonian paths, that is to say, a directed path containing vertices from each partite set, in multipartite tournaments. They established that every 4-strong multipartite tournament is strongly quasi-Hamiltonian-connected–i.e., for every pair of vertices x1, x2, there is a quasi-Hamiltonian path from x1 to x2.In this paper, we characterize all weakly quasi-Hamiltonian-connected multipartite tournaments–i.e., for every pair of vertices, there is at least one quasi-Hamiltonian path between them. Our results include and extend corresponding ones concerning tournaments due to Thomassen.

Cycles through an arc in regular 3-partite tournaments

A c-partite tournament is an orientation of a complete c-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament D is contained in an m-, (m+1)- or (m+2)-cycle for each m∈{3,4,…,|V(D)|−2}, and he also proved this conjecture for m=3,4,5. In 2012, Xu et al. proved that every arc of a regular 3-partite tournament is contained in a 5- or 6-cycle, and in the same paper, the authors also posed the following conjecture:Conjecture 1. If D is an r-regular 3-partite tournament with r≥2, then every arc of D is contained in a 3k- or (3k+1)-cycle for k=1,2,…,r−1.It is known that Conjecture 1 is true for k=1. In this paper, we prove Conjecture 1 for k=2, which implies that Volkmann’s conjecture for m=6 is correct.

Quasi-Hamiltonian cycles in [formula omitted]-strong multipartite tournaments

A c-partite tournament is an orientation of a complete c-partite graph. Recently, M. Lu, et al., introduced the concept of quasi-Hamiltonian cycles, that is to say, cycles containing vertices from each partite set, in multipartite tournaments. W.D. Goddard and O.R. Oellermann established that every strong multipartite tournament contains a quasi-Hamiltonian cycle.In this paper, we show that every k-strong (or k-arc-strong) multipartite tournament contains at least k quasi-Hamiltonian cycles. To that end, we prove the following stronger result: Every strong multipartite tournament contains a vertex whose all out-arcs are contained in a quasi-Hamiltonian cycle. Our results include and extend corresponding ones concerning tournaments due to C. Thomassen, as well as M. Goldberg and J.W. Moon.

Complementary cycles in almost regular multipartite tournaments, where one cycle has length four

Let D be a digraph with vertex set V(D) and independence number α(D). If x∈V(D), then the numbers d+(x) and d−(x) are the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D)=max{max(d+(x),d−(x))−min(d+(y),d−(y))∣x,y∈V(D)}. If ig(D)=0, then D is regular, and if ig(D)≤1, then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph.In 1999, Yeo conjectured that each regular c-partite tournament D with c≥4 and |V(D)|≥8 contains a pair of vertex-disjoint directed cycles of lengths 4 and |V(D)|−4. In 2004, Volkmann confirmed this conjecture for c≥5 and c=4 and α(D)≥4. As a supplement to this result, we prove in this paper the following theorem.Let D be an almost regular c-partite tournament with |V(D)|≥8 such that all partite sets have the same cardinality r. If c≥5 or c=4 and r≥6, then D contains a pair of vertex-disjoint directed cycles of lengths 4 and |V(D)|−4.

Weakly quasi-Hamiltonian-set-connected multipartite tournaments

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. Lu and Guo (submitted for publication) [3] recently introduced strong quasi-Hamiltonian-connectivity of a multipartite tournament D as follows: For any two distinct vertices x and y of D, there is a path with at least one vertex from each partite set of D from x to y and from y to x. We obtain the definition for weak quasi-Hamiltonian-connectivity, where only one of those paths, and weak quasi-Hamiltonian-set-connectivity, where only one such path between every two distinct partite sets has to exist, in a natural way.In this paper, we characterize weakly quasi-Hamiltonian-set-connected multipartite tournaments which extends a result of Thomassen (1980) [6].

Strong subtournaments of order [formula omitted] containing a given vertex in regular [formula omitted]-partite tournaments with [formula omitted

Volkmann and Winzen [L. Volkmann, S. Winzen, Strong subtournaments containing a given vertex in regular multipartite tournaments, Discrete Math. 308 (2008) 5516–5521] showed that each vertex of a regular c-partite tournament with c≥7 partite sets is contained in a strong subtournament of order p for every p∈{3,4,…,c−4}, and conjectured that each vertex of a regular c-partite tournament with c≥5 partite sets is contained in a strong subtournament of order p for every p∈{3,4,…,c}. In this paper we confirm the conjecture when c≥16.

Weakly Cycle Complementary 3-Partite Tournaments

The vertex set of a digraph D is denoted by V(D). A c-partite tournament is an orientation of a complete c-partite graph. Let V 1, V 2, . . . ,V c be the partite sets of D. If there exist two vertex disjoint cycles C and C′ in D such that $${V_{\mathrm{i}}\cap(V(C)\cup V(C'))\neq\emptyset}$$ for all i = 1, 2, . . . , c, then D is weakly cycle complementary. In 2008, Volkmann and Winzen gave the above definition of weakly complementary cycles and proved that all 3-connected c-partite tournaments with c ≥ 3 are weakly cycle complementary. In this paper, we characterize multipartite tournaments are weakly cycle complementary. Especially, we show that all 2-connected 3-partite tournaments that are weakly cycle complementary, unless D is isomorphic to D 3,2, D 3,2,2 or D 3,3,1.

Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ( V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ( V (C2) contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid (4) in 1985 and Z. Song (5) in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and jV (T)j i t for all 3 • tjV (T)j=2. Recently, Volkmann (8) proved that each regular multipartite tournament D of order jV (D)j ‚ 8 is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with c ‚ 3 that are weakly cycle complementary. 1. Terminology In this paper all digraphs are flnite without loops and multiple arcs. The vertex set and the arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x ! y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X ! Y. Furthermore, X ; Y denotes the fact that there is no arc leading from Y to X. If D is a digraph, then the out-neighborhood N + D (x) = N + (x) of a vertex x is the set of vertices dominated by x and the in-neighborhood N i D (x) = N i (x) is the set of vertices dominating x. Therefore, if the arc xy 2 E(D) exists, then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d + (x) = d + (x) = jN + (x)j and d i (x) = d i (x) = jN i (x)j are called the outdegree and the indegree of x, respectively. Furthermore, the numbers - + D = - + = minfd + (x)jx 2 V (D)g and - i D = - i = minfd i (x)jx 2 V (D)g are the minimum outdegree and the minimum

CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we will examine the existence of cycles with r−2 vertices from each partite set in regular multipartite tournaments where the r−2 vertices are chosen arbitrarily. Let D be a regular c-partite tournament and let X ⊆ V (D) be an arbitrary set with exactly 2 vertices of each partite set. For all c ≥ 4 we will determine the minimal value g(c) such that D−X is Hamiltonian for every regular multipartite tournament with r ≥ g(c). 1. Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and the arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x → y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X A Y denotes the fact that there is no arc leading from Y to X. For the number of arcs from X to Y we write d(X, Y ). If D is a digraph, then the out-neighborhood N D (x) = N (x) of a vertex x is the set of vertices dominated by x and the in-neighborhood N− D (x) = N −(x) is the set of vertices dominating x. Therefore, if xy ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d+D(x) = d(x) = |N+(x)| and dD(x) = d−(x) = |N−(x)| are called the outdegree and the indegree of x, respectively. Furthermore, the numbers δ D = δ + = Received December 19, 2005. 2000 Mathematics Subject Classification. 05C20.

Multipartite tournaments: A survey

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. In this survey we mainly describe results on directed cycles and paths in strongly connected c-partite tournaments for c ⩾ 3 . In addition, we include about 40 open problems and conjectures.

Paths and cycles containing given arcs, in close to regular multipartite tournaments

The global irregularity of a digraph D is defined by i g ( D ) = max { d + ( x ) , d − ( x ) } − min { d + ( y ) , d − ( y ) } over all vertices x and y of D (including x = y ). In this paper we prove that if D is a c-partite tournament such that c ⩾ 4 and | V ( D ) | > 476 i g ( D ) + 13 917 then there exists a path of length l between any two given vertices for all 42 ⩽ l ⩽ | V ( D ) | − 1 . There are many consequences of this result. For example we show that all sufficiently large regular c-partite tournaments with c ⩾ 4 have a Hamilton cycle through any given arc, and the condition c ⩾ 4 is best possible. Sufficient conditions are furthermore given for when a c-partite tournament with c ⩾ 4 has a Hamilton cycle containing a given path or a set of given arcs. We show that all sufficiently large c-partite tournaments with c ⩾ 5 and bounded i g are vertex-pancyclic and all sufficiently large regular 4-partite tournaments are vertex-pancyclic. Finally we give a lower bound on the number of Hamilton cycles in a c-partite tournament with c ⩾ 4 .

Cycles with a given number of vertices from each partite set in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d +(x) and d −(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p ∈ ℕ such that d +(x) = d −(x) = p for all vertices x of D.A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether c-partite tournaments with r vertices in each partite set contain a cycle with exactly r − 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c ⩾ 3, then we will show that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a cycle with exactly r − 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2.

Paths with a given number of vertices from each partite set in regular multipartite tournaments

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. For c ⩾ 2 we prove that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a directed path with exactly two vertices from each partite set. Furthermore, if c ⩾ 4 , then we will show that almost all regular c-partite tournaments D contain a directed path with exactly r - s vertices from each partite set for each given integer s ∈ N , if r is the cardinality of each partite set of D. Some related results are also presented.

On cycles in regular 3-partite tournaments

The vertex set of a digraph D is denoted by V ( D ) . A c-partite tournament is an orientation of a complete c-partite graph. In 1991, Jian-zhong Wang conjectured that every arc of a regular 3-partite tournament D is contained in directed cycles of all lengths 3 , 6 , 9 , … , | V ( D ) | . This conjecture is not valid, because for each integer t with 3 ⩽ t ⩽ | V ( D ) | , there exists an infinite family of regular 3-partite tournaments D such that at least one arc of D is not contained in a directed cycle of length t. In this paper, we prove that every arc of a regular 3-partite tournament with at least nine vertices is contained in a directed cycle of length m, m + 1 , or m + 2 for 3 ⩽ m ⩽ 5 , and we conjecture that every arc of a regular 3-partite tournament is contained in a directed cycle of length m, ( m + 1 ) , or ( m + 2 ) for each m ∈ { 3 , 4 , … , | V ( D ) | - 2 } . It is known that every regular 3-partite tournament D with at least six vertices contains directed cycles of lengths 3, | V ( D ) | - 3 , and | V ( D ) | . We show that every regular 3-partite tournament D with at least six vertices also has a directed cycle of length 6, and we conjecture that each such 3-partite tournament contains cycles of all lengths 3 , 6 , 9 , … , | V ( D ) | .

On the connectivity of close to regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d + ( x ) and d - ( x ) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by i g ( D ) = max { d + ( x ) , d - ( x ) } - min { d + ( y ) , d - ( y ) } over all vertices x and y of D (including x = y ) and the local irregularity of a digraph D is i l ( D ) = max | d + ( x ) - d - ( x ) | over all vertices x of D. Clearly, i l ( D ) ⩽ i g ( D ) . If i g ( D ) = 0 , then D is regular and if i g ( D ) ⩽ 1 , then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Let V 1 , V 2 , … , V c be the partite sets of a c-partite tournament such that | V 1 | ⩽ | V 2 | ⩽ ⋯ ⩽ | V c | . In 1998, Yeo proved κ ( D ) ⩾ | V ( D ) | - | V c | - 2 i l ( D ) 3 for each c-partite tournament D, where κ ( D ) is the connectivity of D. Using Yeo's proof, we will present the structure of those multipartite tournaments, which fulfill the last inequality with equality. These investigations yield the better bound κ ( D ) ⩾ | V ( D ) | - | V c | - 2 i l ( D ) + 1 3 in the case that | V c | is odd. Especially, we obtain a 1980 result by Thomassen for tournaments of arbitrary (global) irregularity. Furthermore, we will give a shorter proof of the recent result of Volkmann that κ ( D ) ⩾ | V ( D ) | - | V c | + 1 3 for all regular multipartite tournaments with exception of a well-determined family of regular ( 3 q + 1 ) -partite tournaments. Finally we will characterize all almost regular tournaments with this property.

Almost regular multipartite tournaments containing a Hamiltonian path through a given arc

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d +( x) and d −( x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by i g ( D)=max{ d +( x), d −( x)}−min{ d +( y), d −( y)} over all vertices x and y of D (including x= y). If i g ( D)=0, then D is regular and if i g ( D)⩽1, then D is called almost regular. Recently, Volkmann and Yeo have proved that every arc of a regular multipartite tournament is contained in a directed Hamiltonian path. If c⩾4, then this result remains true for almost all c-partite tournaments D of a given constant irregularity i g ( D). For the case that i g ( D)=1 we will give a more detailed analysis. If c=3, then there exist infinite families of such digraphs, which have an arc that is not contained in a directed Hamiltonian path of D. Nevertheless, we will present an interesting sufficient condition for an almost regular 3-partite tournament D with the property that a given arc is contained in a Hamiltonian path of D.