Regular Multipartite Tournaments Research Articles

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.

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.

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

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 1999, Yeo conjectured that each regular c -partite tournament D with c ≥ 4 and | V ( D ) | ≥ 10 contains a pair of vertex disjoint directed cycles of lengths 5 and | V ( D ) | − 5 . In this paper we shall confirm this conjecture using a computer program for some cases.

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

Strong subtournaments containing a given vertex 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. The global irregularity of a digraph D is defined by i g ( D ) = max x ∈ V ( D ) { d + ( x ) , d − ( x ) } − min y ∈ V ( D ) { d + ( y ) , d − ( y ) } . If i g ( D ) = 0 , then D is regular and if i g ( D ) ≤ 1 , then D is called almost regular. A c -partite tournament is an orientation of a complete c -partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c -partite tournaments contain a strong subtournament of order c when c ≥ 5 , Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c -partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ { 3 , 4 , … , c } . In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ { 3 , 4 , … , c − 4 } .

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.

Strongly [formula omitted]-path-connectivity in almost regular multipartite tournaments

If x is a vertex of a digraph D , 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 almost regular. A digraph D is said to be strongly k-path-connected if for any two vertices x , y ∈ V ( D ) there is an ( x , y ) -path of order k and a ( y , x ) -path of order k in D . In this paper we show that an almost regular c -partite tournament with c ⩾ 8 is strongly 4 -path-connected. Examples show that the condition c ⩾ 8 is best possible.

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.

Longest cycles in almost regular 3-partite tournaments

If D is a digraph, then we denote by V ( D ) its vertex set. A multipartite or c -partite tournament is an orientation of a complete c -partite graph. The global irregularity of a digraph D is defined by i g ( D ) = max { max ( d + ( x ) , d - ( x ) ) - min ( d + ( y ) , d - ( y ) ) | x , y ∈ V ( D ) } . If i g ( D ) = 0 , then D is regular, and if i g ( D ) ⩽ 1 , then D is called almost regular. In 1997, Yeo has shown that each regular multipartite tournament is Hamiltonian. This remains valid for almost all almost regular c -partite tournaments with c ≥ 4 . However, there exist infinite families of almost regular 3-partite tournaments without any Hamiltonian cycle. In this paper we will prove that every vertex of an almost regular 3-partite tournament D is contained in a directed cycle of length at least | V ( D ) | - 2 . Examples will show that this result is best possible.

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 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.

Cycles through a given arc and certain partite sets in almost 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 ). 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. In 1998, Guo and Kwak showed that, if D is a regular c -partite tournament with c ⩾4, then every arc of D is in a directed cycle, which contains vertices from exactly m partite sets for all m ∈{4,5,…, c }. In this paper we shall extend this theorem to almost regular c -partite tournaments, which have at least two vertices in each partite set. An example will show that there are almost regular c -partite tournaments with arbitrary large c such that not all arcs are in directed cycles through exactly 3 partite sets. Another example will show that the result is not valid for the case that c =4 and there is only one vertex in a partite set.

Cycles through a given arc and certain partite sets in almost 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). 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. In 1998, Guo and Kwak showed that, if D is a regular c-partite tournament with c⩾4, then every arc of D is in a directed cycle, which contains vertices from exactly m partite sets for all m∈{4,5,…, c}. In this paper we shall extend this theorem to almost regular c-partite tournaments, which have at least two vertices in each partite set. An example will show that there are almost regular c-partite tournaments with arbitrary large c such that not all arcs are in directed cycles through exactly 3 partite sets. Another example will show that the result is not valid for the case that c=4 and there is only one vertex in a partite set.

Hamiltonian paths, containing a given path or collection of arcs, in close to 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. If x is a vertex of a digraph D, then we denote by d +( x) and d −( x) the outdegree and 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 called regular. Let V 1, V 2,…, V c be the partite sets of a c-partite tournament D such that | V 1|⩽| V 2|⩽⋯⩽| V c |. If P is a directed path of length q in the c-partite tournament D such that |V(D)|⩾2i g(D)+3q+2|V c|+|V c−1|−2, then we prove in this paper that there exists a Hamiltonian path in D, starting with the path P. Examples will show that this condition is best possible. As an application of this theorem, we prove that each arc of a regular multipartite tournament is contained in a Hamiltonian path. Some related results are also presented.

All regular multipartite tournaments that are cycle complementary

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. A digraph D is cycle complementary if there exist two vertex-disjoint directed cycles spanning the vertex set V( D) of D. In this paper we prove that each regular c-partite tournament D of order | V( D)|⩾6 with c⩾3 is cycle complementary, unless D is isomorphic to T 7 or to D 3,2, where T 7 is a 3-regular tournament of order 7, and D 3,2 is a 2-regular 3-partite tournament such that there are exactly two vertices in each partite set.

The cycle structure of regular multipartite tournaments

A multipartite tournament is an orientation of a complete multipartite graph. A tournament is a multipartite tournament, each partite set of which contains exactly one vertex. Alspach (Canad. Math. Bull. 10 (1967) 283) proved that every regular tournament is arc-pancyclic. Although all partite sets of a regular multipartite tournament have the same cardinality, Alspach's theorem is not valid for regular multipartite tournaments. In this paper, we prove that if the cardinality common to all partite sets of a regular n-partite ( n⩾3) tournament T is odd, then every arc of T is in a cycle that contains vertices from exactly m partite sets for all m∈{3,4,…, n}. This result extends Alspach's theorem for regular tournaments to regular multipartite tournaments. We also examine the structure of cycles through arcs in regular multipartite tournaments.

Almost all almost regular c-partite tournaments with c⩾5 are vertex pancyclic

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d +( x) be the outdegree and d −( x) the indegree of the vertex x in D. The minimum (maximum) outdegree and the minimum (maximum) indegree of D are denoted by δ + ( Δ +) and δ − ( Δ −), respectively. In addition, we define δ=min{ δ +, δ −} and Δ=max{ Δ +, Δ −}. A digraph is regular when δ= Δ and almost regular when Δ− δ⩽1. Recently, the third author proved that all regular c-partite tournaments are vertex pancyclic when c⩾5, and that all, except possibly a finite number, regular 4-partite tournaments are vertex pancyclic. Clearly, in a regular multipartite tournament, each partite set has the same cardinality. As a supplement of Yeo's result we prove first that an almost regular c-partite tournament with c⩾5 is vertex pancyclic, if all partite sets have the same cardinality. Second, we show that all almost regular c-partite tournaments are vertex pancyclic when c⩾8, and third that all, except possibly a finite number, almost regular c-partite tournaments are vertex pancyclic when c⩾5.

The cycle structure of regular multipartite tournaments

The cycle structure of regular multipartite tournaments