Strong subtournaments of multipartite tournaments

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d(x) the indegree of the vertex x in D. The minimum (maximum) out degree and the minimum (maximum) in degree of D are denoted by J+ (~+) and J(~-), respectively. Furthermore, we define J = mini J+, J-} and ~ = maxi ~ + , ~ -}. A digraph D is almost regular, if ~ 8 ::; 1. If Vi, Vz, ... , Vn are the partite sets of an n-partite tournament D, then we define "((D) = minl:s;i$n{IYiI}. In this paper we prove that every almost regular n-partite tournament with n 2:: 4 contains a strongly connected subtournament of order p for each p E {3, 4, ... ,;z, I}. Examples show that this result is best possible for n = 4. If in addition, "((D) < 3n/2 6, for an almost regular n-partite tournament D with n 2:: 5, then D even contains a strong subtournament of order n. 1. Terminology and Introduction An n-partite or multipartite tournament is an orientation of a complete n-partite graph, and a tournament is an n-partite tournament with exactly n vertices. The vertex set of a digraph D is denoted by V(D) and the arc set by A(D). The number IV (D) I is called the order of the digraph D. If there is an arc from x to y in a digraph D, then we say that x dominates y, denoted by x -+ y. Let X and Y be two disjoint subsets of V(D). We use X -+ Y to denote the fact that x -7 y for all vertices x E X and all y E Y. Furthermore, if x -+ y for all x E X and y E Y, which are in different partite sets of a multipartite tournament, then we write X ~ Y. By d(X, Y) we denote the number of arcs from the set X to the Australasian Journal of Combinatorics 20(1999), pp.189-196 set Y, i.e., d(X, Y) = I{xy E A(D) : x E X, Y E Y}I. The vertex x is a neighbor of the vertex y, if x -+ y or y -+ x. The outset N+(.1J, D) = N+(x) of a vertex x in D is the set of vertices dominated by x, and the inset N(x, D) = N(x) is the set of vertices dominating x. We denote by d+ D) d+ (x) = I N+ (x) I the out degree and by d-(x, D) = d-(x) = IN-(x)1 the indegree of the vertex x E V(D). The minimum (maximum) out degree and the minimum (maximum) indegree of D are denoted by 5+(D) (6.+(D)) and 5-(D) (6.-(D)), respectively. In addition, we define 5(D) = min{ 5+(D), 5-(D)} and 6.(D) = max{6. +(D), 6. -(D)}. A digraph D is regular, if 5 ( D) = 6. ( D) and almost regular, if 6. ( D) 5 ( D) :::; 1. For a vertex set X of D, we define D[X] as the sub digraph induced by X. a (path) we mean a directed cycle (directed path). A cycle (path) of a digraph D is Hamiltonian if it includes all the vertices of D. A digraph D is said to be strongly connected or just strong, if for every pair x, y of vertices in D, there is a path from x to y. A strong component of D is a maximal induced strong sub digraph of D. If D is an n-partite tournament with the partite sets VI, V2 , ..• , Vn such that IVII :::; IV2 1 :::; ... :::; I, then IVnl a(D) is the independence number of D, and we define by ,(D) = IViI. In 1976, Bondy [2] has proved that a strongly connected n-partite tournament contains an m-cycle for all m between 3 and n. If one could find in such an n-partite tournament a strong subtournament of order n, then by the well-known theorem of Moon (see Theorem 2.1 below), Bondy's result would be a diryct consequence. But the next example will show that this way is not practicable in general. Example 1.1 Let Vi, V2 , ..• , Vn be the partite sets of an n-partite tournament with n ;:::: 4 and Vn UI U U2 U ... U UnI such that Vi -+ Vi for 1 :::; i < j :::; n 1, {Vi, 1/2, ... , vt-I, vt+d -+ Ut , and Ut -+ {vt, vt+2, vt+3,"" Vn-d for 1 :::; t :::; n l. Then it is a simple matter to verify that the resulting n-partite tournament D is strongly connected. But the largest strong sub tournament of D only consists of three vertices. There is extensive literature on cycles and paths in multipartite tournaments, see e.g., Bang-Jensen and Gutin [1], Guo [3], Gutin [4], Volkmann [7], and Yeo [8]. In view of this, it is somewhat surprising that the closely-related question for strongly connected subtournaments in multipartite tournaments have, as yet, received no attention. In this paper we will develop the first contributions to this interesting problem. We prove that every almost regular n-partite tournament with n ;:::: 4 contains a strongly connected subtournament of Qrder p for each p E {3, 4, ... ,n I}. An infinite family of regular 4-partite tournaments without a strong subtournament of order 4 shows that this result is best possible for n = 4. If in addition, ,(D) < 3n/2 6,' for an almost regular n-partite tournament D with n ;:::: 5, then we are able to show that D even contains a strong sub tournament of order n. In regular n-partite tournaments one can weaken the last condition slightly to ,( D) < 3n/2 2. But since we are quite sure that it is possible to extend the last two results, we omit the proofs.