On the cycle structure of in-tournaments

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Therefore, in-tournaments are a generalization of local tournaments where, for every vertex, the set of inneighbors as well as the set of out-neighbors induce a tournament. While local tournaments have been intensively studied very little is known about in-tournaments. It is the purpose of this paper to give more information about in-tournaments where we will focus mainly on the cycle structure of these digraphs. We will investigate the extendability of cycles and the influence of the minimum indegree on the cycle structure. In particular, we show that every strong in-tournament of order n with minimum indegree at least ~ is pancyclic. 1 Terminology and Introduction Throughout this paper we will consider digraphs that contain no multiple arcs, no loops and no cycles of length 2. We call these digraphs oriented graphs. An intournament is an oriented graph such that the set of negative neighbors of every vertex induces a tournament, i.e. every pair of distinct vertices that have a common positive neighbor are connected by exactly one arc. A digraph D is determined by its set of vertices and its set of arcs, denoted by V(D) and E(D), respectively. We call D a connected digraph if the underlying graph is connected. For xy E E(D) where x, y E V(D), we write x -t y and we say that x dominates y or y i8 dominated by x. Furthermore, y is a positive neighbor or out-neighbor of x and x is a negative neighbor or in-neighbor of y. Let 51 and 52 be disjoint subsets of V(D). If 81 -t 82 for every 81 E 51 and 82 E 52 we denote this by 51 -t 52' If 52 = {y} then we use 51 -t y instead of 8 1 -t {y}. For 5 ~ V(D), the digraph which is induced by the vertices of 5 is denoted by D[8]. The outdegree d+(x, 8) and indegree d-(x, 8) with respect to 5 of a vertex x E V(D) are defined to be the number of positive and negative neighbors of x in 8, respectively. In the case when 8 = V (D), we also write d+ (x) and d(x). Australasian Journal of Combinatoric;s 18(1998). pp.293-301 The minimum outdegree 5+(D) and the minimum indegree 5-(D) of D are given by min { d+ (x) I x E V (D)} and min { d(x) I x E V (D)}, respectively. Furthermore, 5(D) = min{ 5+(D), 5-(D)} is the minimum degree of D. Analogously, the maximum outdegree of D is defined as ,6.+ (D) = max { d+ (x) I x E V (D)}. If d+ (x) = d(x) = p for every x E V (D), then D is called p-regular. All cycles and paths mentioned here are oriented cycles and oriented paths. A Hamiltonian path of a digraph D is a path that consists of all the vertices of D. Analogously, a if amiltonian cycle is a cycle containing all the vertices of D. Tha-length of a shortest cycle in a digraph D is called the girth of D, denoted by g(D). A cycle C in D is extendable if D contains a cycle C f such that V (G) c V (G f ) and IV(Cf)1 = IV(G)I + 1. We call C to be a k-cycle if C consists of k vertices. A digraph D of order n is called pancyclic if D contains a k-cycle for every 3 :::; k :::; n. If every vertex of D belongs to a cycle of length k for every 3 ~ k ~ n, then D is called vertex pan cyclic. The study of tournaments and their different generalizations is one of the most attractive subjects in the work on digraphs. One type of generalization transfers the adjacency between every pair of distinct vertices in tournaments to only those pairs where both vertices belong either to the positive or to the negative neighborhood of some vertex of the digraph. This leads to the class of local tournaments, or more generally, to the class of locally semi complete digraphs where adjacent vertices may be connected by two mutually opposite arcs. The research about the structure of these digraphs evolved into a very productive area. In particular, the Ph. D. theses of Y. Guo (5] and J. Huang (8] have been devoted to this subject. As a generalization of local tournaments, J. Bang-Jensen, J. Huang, and E. Prisner [1] studied the class of in-tournaments, where only the set of in-neighbors of every vertex induces a tournament. But very lIttle work has been done concerning in-tournaments and it is the purpose of this paper to give more information about the properties of this family of digraphs. We focus on the cycle structure of intournaments where we consider the extendability of cycles and the influence of the minimum indegree on the cycle structure. In particular, we show that every strong in-tournament of order n with minimum in degree at least ~ is pancyclic. 2 Preliminary results In this section we will state some known results which either will be useful in our investigations or will be generalized later on. The first two are due to Redei [10] and Moon [9], respectively, and deal with the structure of tournaments. Theorem 2.1 Every tournament contains a Hamiltonian path. Theorem 2.2 Every strong tournament is vertex pancyclic. The next two results on long cycles in strong in-tournaments were found by BangJensen, Huang, and Prisner [1].