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.     

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.     

Every cycle-connected multipartite tournament has a universal arc

A digraph D is cycle-connected if for every pair of vertices u,v∈V(D) there exists a directed cycle in D containing both u and v. In 1999, Ádám [A. Ádám, On some cyclic connectivity properties of directed graphs, Acta Cybernet. 14 (1) (1999) 1-12] posed the following problem. Let D be a cycle-connected digraph. Does there exist a universal arc in D, i.e., an arc e∈A(D) such that for every vertex w∈V(D) there is a directed cycle in D containing both e and w?A c-partite or multipartite tournament is an orientation of a complete c-partite graph. Recently, Hubenko [A. Hubenko, On a cyclic connectivity property of directed graphs, Discrete Math. 308 (2008) 1018-1024] proved that each cycle-connected bipartite tournament has a universal arc. As an extension of this result, we show in this note that each cycle-connected multipartite tournament has a universal arc.     

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 ). 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 . The last one is about two disjoint cycles passing through all partite sets.     

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

     

Outpaths of arcs in multipartite tournaments

1. IntroductionThroughout the paPer, we use the terminology and notation of [1] and [2]. Let D =(V(D), A(D)) be a digraPh. If xy is an arc of a digraPh D, then we say that x dominatesy, denoted by x - y. More generally, if A and B are two disjoint vertex sets of D such thatevery vertex of A dominates every vertex of B, then we say that A dominates B, denotedby A - B. The outset N (x) of a vertex x is the set of vertices dominated by x in D,and the inset N--(x) is the set of vertices d…     

The structure of 4‐strong tournaments containing exactly three out‐arc pancyclic vertices

Yao et al. (Discrete Appl Math 99 (2000), 245–249) proved that every strong tournament contains a vertex u such that every out‐arc of u is pancyclic and conjectured that every k‐strong tournament contains k such vertices. At present, it is known that this conjecture is true for k = 1, 2, 3 and not true for k?4. In this article, we obtain a sufficient and necessary condition for a 4‐strong tournament to contain exactly three out‐arc pancyclic vertices, which shows that a 4‐strong tournament contains at least four out‐arc pancyclic vertices except for a given class of tournaments. Furthermore, our proof yields a polynomial algorithm to decide if a 4‐strong tournament has exactly three out‐arc pancyclic vertices.     

Extendable Cycles in Multipartite Tournaments

An n-partite tournament is an orientation of a complete n-partite graph. If D is a strongly connected n-partite (n3) tournament, then we shall prove that every partite set of D has at least one vertex which lies on a cycle C_m of each length m for such that V(C₃)V(C₄)V(C_n), where V(C_m) is the vertex set of C_m for . This result extends those of Bondy [2], Guo and Volkmann [4], Gutin [6], Moon [8], and Yeo [12].Final version received: June 9, 2003     

On arc-traceable local tournaments

A digraph D is arc-traceable if for every arc xy of D, the arc xy belongs to a directed Hamiltonian path of D. A local tournament is an oriented graph such that the negative neighborhood as well as the positive neighborhood of every vertex induces a tournament. It is well known that every tournament contains a directed Hamiltonian path and, in 1990, Bang-Jensen showed the same for connected local tournaments. In 2006, Busch, Jacobson and Reid studied the structure of tournaments that are not arc-traceable and consequently gave various sufficient conditions for tournaments to be arc-traceable. Inspired by the article of Busch, Jacobson and Reid, we develop in this paper the structure necessary for a local tournament to be not arc-traceable. Using this structure, we give sufficient conditions for a local tournament to be arc-traceable and we present examples showing that these conditions are best possible.     

Toppling kings in multipartite tournaments by introducing new kings

Let and be two n-tuples of nonnegative integers. An all-4-kings n-partite tournament T(V₁,V₂,…V_n) is said to have a -property if there exists an n-partite tournament T₁(W₁,W₂,…,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 T₁;

(3)

exactly c_i 4-kings of W_i are not vertices of V_i.

We describe all pairs such that there exists an n-partite tournament having -property.     

On Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n≥4, then each arc of T lies on a cycle whose vertices are from exactly κ partite sets for κ=4,5,…,n. Our result, in a sense, generalizes a theorem due to Alspach.     

Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in‐tournaments

A directed cycle C of a digraph D is extendable if there exists a directed cycle C′ in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non‐Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3≤k<n: a digraph D is k ‐extendable if every directed cycle of length t, where k≤t<n, is extendable. Moreover, D is called fully k ‐extendable if D is k ‐extendable and every vertex of D belongs to a directed cycle of length k. An in‐tournament is an oriented graph such that the in‐neighborhood of every vertex induces a tournament. This class of digraphs which generalizes the class of tournaments was introduced by Bang‐Jensen, Huang and Prisner in 1993. Tewes and Volkmann showed that every connected in‐tournament D of order n with minimum degree δ≥1 is ( ) ‐extendable. Furthermore, if D is a strongly connected in‐tournament of order n with minimum degree δ=2 or , then D is fully ( ) ‐extendable. In this article we shall see that if , every vertex of D belongs to a directed cycle of length , which means that D is fully ( ) ‐extendable. This confirms a conjecture of Tewes and Volkmann. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 82–92, 2010     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.     

Ranking the vertices of an r-partite paired comparison digraph

A paired comparison digraph D is a weighted digraph in which the sum of the weights of arcs, if any, joining two vertices is exactly one. A one-to-one mapping from V(D) onto {1,2…,|V(D)|} is called a ranking of D, and a ranking α of D is optimal if the backward length of α is minimum. We say that D is r-partite if V(D) can be partitioned into V₁∪…∪V_r so that every arc of D joins a vertex of V_i to a vertex of V_i, where i ≠ j. We show that we can easily obtain all the optimal rankings of a certain r-partite paired comparison digraph.     

Every Cycle-Connected Multipartite Tournament with δ ≥ 2 Contains At Least Two Universal Arcs

A digraph D = (V(D), A(D)) is called cycle-connected if for every pair of vertices ${u, v\in V(D)}$ there exists a cycle containing both u and v. Ádám (Acta Cybernet 14(1):1–12, 1999) proposed the question: Let D be a cycle-connected digraph. Does there exist a universal arc in D, i.e., an arc ${e\in A(D)}$ such that for every vertex ${w\in V(D)}$ there exists a cycle C in D containing both e and w?. Recently, Lutz Volkmann and Stefan Winzen have proved that this conjecture is true for multipartite tournaments. As an improvement of this result, we show in this note that every cycle-connected multipartite tournament with δ ≥ 2 contains at least two universal arcs.     

Cycle extendability in graphs and digraphs

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.     

A degree sum condition for long cycles passing through a linear forest

Let G be a (k+m)-connected graph and F be a linear forest in G such that |E(F)|=m and F has at most k-2 components of order 1, where k?2 and m?0. In this paper, we prove that if every independent set S of G with |S|=k+1 contains two vertices whose degree sum is at least d, then G has a cycle C of length at least min{d-m,|V(G)|} which contains all the vertices and edges of F.     

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.     

Solution of a conjecture of Volkmann on longest paths through an arc in strongly connected in‐tournaments

An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. Let m = 4 or m = 5 and let D be a strongly connected in‐tournament of order such that each arc belongs to a directed path of order at least m. In 2000, Volkmann showed that if D contains an arc e such that the longest directed path through e consists of exactly m vertices, then e is the only arc of D with that property. In this article we shall see that this proposition is true for , thereby validating a conjecture of Volkmann. Furthermore, we prove that if we ease the restrictions on the order of D to , the in‐tournament D in question has at most two such arcs. In doing so, we also give a characterization of the in‐tournaments with exactly two such arcs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 130–148, 2009