On multipartite tournaments

Journal of Graph Theory, 2020

A digraph D with n vertices is Hamiltonian (pancyclic, vertex-pancyclic, respectively) if D contains a Hamilton cycle (a cycle of every length 3, 4,. .. n, for every vertex v ∈ V (D), a cycle of every length 3, 4,. .. n through v, respectively.) It is well-known that a strongly connected tournament is Hamiltonian (Camion 1959), pancyclic (Harary and Moser 1966), and vertex pancyclic (Moon 1968). A digraph D is cycle extendable if for every non-Hamiltonian cycle C of D, there is a cycle C such that C contains all vertices of C plus another vertex of D. A cycle extendable digraph is fully cycle extendable if for every vertex v ∈ V (D), there exists a cycle of length 3 through v. Note that full cycle extendability is a stronger property than vertex pancyclicity. Hendry (1989) showed that not every strongly connected tournament is fully cycle extendable and characterized an infinite wide class of strongly connected tournaments, which are not fully cycle extendable. A k-partite tournament is an orientation of a k-partite complete graph (for k = 2, it is called a bipartite tournament). Gutin (1984) and Häggkvist and Manoussakis (1989) characterized Hamiltonian bipartite tournaments. A bipartite digraph D with n vertices is even pancyclic (even vertex pancyclic, respectively) if D contains a cycle of every even length 4, 6,. .. , n (a cycle of every even length 4, 6,. .. , n through v for every v ∈ V (D), respectively). Beineke and Little (1982) and Zhang (1984) 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 the case of tournaments, every even pancyclic bipartite tournament is fully cycle extendable. We show that this result cannot be extended to k-partite tournaments for any fixed k ≥ 3 (where we naturally replace even vertex pancyclicity by vertex pancyclicity).

2018

Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipartite tournaments.

A graph or digraph is hamiltonian if it contains a cycle that visits every vertex, and traceable if it contains a path that visits every vertex. A (di)graph is k-traceable if each of its induced subdigraphs of order k is traceable. A digraph D is strong if for every pair u, v of vertices in D there is a directed path from u to v and a directed path from v to u.

Journal of Graph Theory, 1998

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

Discrete Mathematics

Let k ≥ 2 be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k − 1 contains k vertex-disjoint cycles. Recently Bai, Li and Li proved this conjecture for bipartite digraphs. In this paper we prove that every bipartite tournament with minimum out-degree at least 2k − 2, minimum in-degree at least 1 and partite sets of cardinality at least 2k contains k vertex-disjoint 4-cycles whenever k ≥ 3. Finally, we show that every bipartite tournament with minimum degree δ = min{δ + , δ − } at least 1.5k − 1 contains at least k vertex-disjoint 4-cycles.

ArXiv, 2020

We show that for each non-negative integer k, every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 7(k - 1).

Canadian mathematical bulletin, 1970

The average connectivity of a digraph is the average, over all ordered pairs of vertices, of the maximum number of internally disjoint directed paths connecting these vertices. Among the results in this paper, we determine the minimum average connectivity among all orientations of the complete multipartite graph K n1 ,n2, ... ,nk and the maximum average connectivity when all partite sets have the same order.

Discrete Mathematics, 2010

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 3free 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.

The collection of convex subsets of a multipartite tournament T forms a lattice C(T ). Given a lattice structure for C(T ), we deduce properties of T . In particular, we find conditions under which we can detect clones in T (i.e. vertices with identical arc orientations). We also determine conditions on the lattice which will imply that T is bipartite, except for a few cases. We classify the ambiguous cases. Finally, we study a property of C(T ) we call the anti-bipartite condition. We prove a result on directed cycles in multipartite tournaments satisfying the anti-bipartite condition, and determine the minimum number of partitions in such multipartite tournaments.