Partite Tournament Research Articles

All Feedback Arc Sets of a Random Turán Tournament Have $\lfloor {n}/{k}\rfloor-{k}+1$ Disjoint ${k}$-Cliques (and This Is Tight)

We look at structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph $H$ and an oriented graph $G$, let $f_H(G)$ be the maximum number of pairwise disjoint copies of $H$ that can be found in {\em all} feedback arc sets of $G$. In particular, to make $G$ acyclic, one must remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Most intriguing is the case where $H$ is a $k$-clique, where the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we determine $f_k(G)$ precisely for almost all $k$-partite tournaments. Let $s(G)$ denote the size of the smallest vertex class of a $k$-partite tournament $G$. We prove that for all sufficiently large $s=s(G)$, a random $k$-partite tournament $G$ satisfies $f_k(G) = s(G)-k+1$ almost surely. In particular, as the title states, $f_k(G) = \lfloor n/k\rfloor-k+1$ almost surely, where $G$ is a random orientation of the Tur\'an graph $T(n,k)$.

The niche graphs of multipartite tournaments

The niche graph of a digraph $D$ has $V(D)$ as the vertex set and an edge $uv$ if and only if $(u,w) \in A(D)$ and $(v,w) \in A(D)$, or $(w,u) \in A(D)$ and $(w,v) \in A(D)$ for some $w \in V(D)$. The notion of niche graph was introduced by Cable et al. (1989) as a variant of competition graph. If a graph is the niche graph of a digraph $D$, it is said to be niche-realizable through $D$. If a graph $G$ is niche-realizable through a $k$-partite tournament for an integer $k \ge 2$, then we say that the pair $(G, k)$ is niche-realizable. Bowser et al. (1999) studied the graphs that are niche-realizable through a tournament and Eoh et al. (2018) studied niche-realizable pairs $(G, k)$ for $k=2$. In this paper, we study niche-realizable pairs $(G, k)$ when $G$ is a graph and $k$ is an integer at least $3$ to extend their work. We show that the niche graph of a $k$-partite tournament has at most three components if $k \ge 3$ and is connected if $k \ge 4$. Then we find all the niche-realizable pairs $(G, k)$ when $G$ is a disconnected graph, when $G$ is a complete graph, and when $G$ is a connected triangle-free graph.

Outpaths of arcs in regular 3-partite tournaments

Outpaths of arcs in regular 3-partite tournaments

Hamiltonicity, pancyclicity, and full cycle extendability in multipartite tournaments

AbstractA digraph with vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if contains a Hamilton cycle (a cycle of every length , for every vertex , a cycle of every length through , respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph is cycle extendable if for every non‐Hamiltonian cycle of , there is a cycle such that contains all vertices of plus another vertex of . A cycle extendable digraph is fully cycle extendable if for every vertex , there exists a cycle of length 3 through . Note that full cycle extendability is a stronger property than vertex pancyclicity. While it is well‐known that every strongly connected tournament is vertex pancyclic, Hendry showed that not every strongly connected tournament is fully cycle extendable and characterized an infinite class of strongly connected tournaments, which are not fully cycle extendable. A ‐partite tournament is an orientation of a ‐partite complete graph (for , it is called a bipartite tournament). Gutin and later Häggkvist and Manoussakis characterized Hamiltonian bipartite tournaments. A bipartite digraph with vertices is even pancyclic (even vertex pancyclic, respectively) if contains a cycle of every even length (a cycle of every even length through for every , respectively). Beineke and Little, and Zhang proved that every bipartite tournament is even pancyclic and even vertex pancyclic, respectively, if and only if it is Hamiltonian and does not belong to a well‐defined infinite class of regular bipartite tournaments. We prove that unlike in the case of tournaments, every even pancyclic bipartite tournament is fully cycle extendable. We show that this result cannot be extended to ‐partite tournaments for any fixed (where we naturally replace even vertex pancyclicity by vertex pancyclicity).

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.

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.

On the vertices of a 3-partite tournament not in triangles

Let T be a 3-partite tournament and F3(T) be the set of vertices of T not in triangles. We prove that, if the global irregularity of T, ig(T), is one and |F3(T)|>3, then F3(T) must be contained in one of the partite sets of T and |F3(T)|≤⌊k+14⌋+1, which implies |F3(T)|≤⌊n+512⌋+1, where k is the size of the largest partite set and n the number of vertices of T. Moreover, we give some upper bounds on the number, as well as results on the structure of said vertices within the digraph, depending on its global irregularity.

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.

Cycles through arcs in multipartite tournaments and a conjecture of Volkmann

Volkmann [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148–1150] conjectured that a strong c -partite tournament with c ≥ 3 contains three arcs that belong to a cycle of length m for each m ∈ { 3 , 4 , … , c } . In this work, we prove that Volkmann’s conjecture is true.

The number of [formula omitted]-free vertices on 3-partite tournaments

Let T be a 3-partite tournament. We say that a vertex v is C 3 ⃗ -free if v does not lie on any directed triangle of T . Let F 3 ( T ) be the set of the C 3 ⃗ -free vertices in a 3-partite tournament and f 3 ( T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F 3 ( T ) must be contained in one of the partite sets of T . It is also shown that for every regular 3-partite tournament, f 3 ( T ) does not exceed n 9 , where n is the order of T . On the other hand, we give an infinite family of strongly connected tournaments having n − 4 C 3 ⃗ -free vertices. Finally we prove that for every c ≥ 3 there exists an infinite family of strongly connected c -partite tournaments, D c ( T ) , with n − c − 1 C 3 ⃗ -free vertices.

Toppling kings in multipartite tournaments by introducing new kings

Let t = ( t 1 , t 2 , … , t n ) and c = ( c 1 , c 2 , … , c n ) be two n -tuples of nonnegative integers. An all-4-kings n -partite tournament T ( V 1 , V 2 , … V n ) is said to have a ( t , c ) - property if there exists an n -partite tournament T 1 ( W 1 , W 2 , … , W n ) such that for each i ∈ { 1 , … , n } : (1) V i ⊆ W i ; (2) exactly t i 4-kings of V i are not 4-kings in T 1 ; (3) exactly c i 4-kings of W i are not vertices of V i . We describe all pairs ( t , c ) such that there exists an n -partite tournament having ( t , c ) -property.

Three supplements to Reid’s theorem in multipartite tournaments

An n -partite tournament is an orientation of a complete n -partite graph. In this paper, we give three supplements to Reid’s theorem [K.B. Reid, Two complementary circuits in two-connected tournaments, Ann. Discrete Math. 27 (1985) 321–334] in multipartite tournaments. The first one is concerned with the lengths of cycles and states as follows: let D be an ( α ( D ) + 1 )-strong n -partite tournament with n ≥ 6 , where α ( D ) is the independence number of D , then D contains two disjoint cycles of lengths 3 and n − 3 , respectively, unless D is isomorphic to the 7-tournament containing no transitive 4-subtournament (denoted by T 7 1 ). The second one is obtained by considering the number of partite sets that cycles pass through: every ( α ( D ) + 1 )-strong n -partite tournament D with n ≥ 6 contains two disjoint cycles which contain vertices from exactly 3 and n − 3 partite sets, respectively, unless it is isomorphic to T 7 1 . The last one is about two disjoint cycles passing through all partite sets.

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.

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 } .

3-Kings in 3-partite tournaments

We present all possible distributions of 3-kings in 3-partite tournaments with at most one transmitter.

Almost regular [formula omitted]-partite tournaments contain a strong subtournament of order [formula omitted] when [formula omitted

An orientation of a complete graph is a tournament, and an orientation of a complete c -partite graph is a c -partite tournament. 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. In 1999, L. Volkmann showed that, if D is an almost regular c -partite tournament with c ⩾ 4 , then D contains a strongly connected subtournament of order p for every p ∈ { 3 , 4 , … , c - 1 } and he conjectured that this also holds for p = c , if c ⩾ 5 . In this paper, we settle this conjecture in affirmative.

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.

A remark on cycles through an arc in strongly connected multipartite tournaments

A multipartite or c -partite tournament is an orientation of a complete c -partite graph. In this note we prove that a strongly connected c -partite tournament with c ≥ 3 contains an arc that belongs to a directed cycle of length m for every m ∈ { 3 , 4 , … , c } . This result extends those of Bondy [J.A. Bondy, Diconnected orientation and a conjecture of Las Vergnas, J. London Math. Soc. 14 (1976) 277–282] and Yeo [A. Yeo, Diregular c -partite tournaments are vertex-pancyclic when c ≥ 5 , J. Graph Theory 32 (1999) 137–152] for multipartite tournaments and Camion [P. Camion, Chemins et circuits hamiltoniens des graphes complets, C. R. Acad. Sci. Paris 249 (1959) 2151–2152] and Harary and Moser [F. Harary, L. Moser, The theory of round robin tournaments, Amer. Math. Monthly 73 (1966) 231–246] for tournaments.

Score sets ink-partite tournaments

The set S of distinct scores (outdegrees) of the vertices of a k- partite tournament T(X1,X2,···,Xk) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite tournament for every n k 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.