Longest paths through an arc in strongly connected in-tournaments

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Recently, we have shown that every arc of a strongly connected tournament of order n is contained in a directed path of order r(n + 3)/21. This is no longer valid for strongly connected in-tournaments, because there exist examples containing an arc with the property that the longest directed path through this arc consists of three vertices. But in this paper we shall see that every strongly connected in-tournament has at most one such arc. More general, we shall prove that if a strongly connected in-tournament D of order n contains m 2 :::; n 3 arcs a3, a4," ., am such that the longest directed path through ak consists of k vertices for 3 :::; k :::; m, then all other arcs of D belong to directed paths of order at least m + 1. Furthermore, we shall show that every arc of a strongly connected in-tournament is contained in a directed path of order k + 2, when max{ 6+,6-} ~ k, where 6+ and 6is the minimum out degree and the minimum indegree, respectively. 1. Terminology and introduction The vertex set and the arc set of a digraph D are denoted by V(D) and E(D), respectively. The number IV(D)I is the order of the digraph D. Throughout this paper we will consider digraphs without multiple arcs, loops, or directed cycles of length two. Such digraphs are called oriented graphs. If there is an arc from x to y in D, then y is a positive neighbor of x and x is a negative neighbor of y, and we also say that x dominates y, denoted by x -+ y. More generally, let A and B be two disjoint sub digraphs of D or subsets of V(D). If x -+ y for every vertex x in A and every vertex y in B, then we write A -+ B and say that A dominates B. Two vertices x and y of a digraph are adjacent when x -+ y or y -+ x. The outset N+(x) of a vertex x is the set of vertices dominated by x, and the inset N(x) is the set of vertices dominating x. The numbers d+(x) = IN+(x)1 and d-(x) = IN-(x)1 are called outdegree and indegree, respectively. The minimum outdegree 8+ and the minimum Australasian Journal of Combinatorics 21(2000). pp.95-106 indegree 6of D are given by min{d+(x) I x E V(D)} and min{d-(x) I x E V(D)}, respectively. For A ~ V(D), we define D[A] as the sub digraph induced by A. By a cycle (path) we mean a directed cycle (directed path). A cycle or a path of order m is called an m-cycle or an m-path, respectively. A cycle (path) of a digraph D is Hamiltonian if it includes all the vertices of D. We speak of a connected digraph if the underlying graph is connected. A digraph D is said to be strongly connected or just strong, if for every pair x, y of vertices of D, there is a path from x to y. A strong component of D is a maximal induced strong subdigraph of D. A digraph D is k-connected if for any set S of at most k 1 vertices, the subdigraph D S is strong. A minimal separating set of a strong digraph D is a subset S c V(D) such that D S is not strong, but D S' is strong for any S' c S. An in-tournament is an oriented graph with the property that the inset of every vertex induces a tournament, i.e., every pair of distinct vertices that have a common positive neighbor are adjacent. A local tournament is an oriented graph such that the inset as well as the outset of every vertex induces a tournament. Throughout this paper all subscripts are taken modulo the corresponding number. Local tournaments were introduced by Bang-Jensen [1] in 1990 and there exists extensive literature on this class of digraphs, e.g., the survey paper of Bang-Jensen and Gutin [2]. In particular, the Ph. D. theses of Y. Guo [4] and J. Huang [5] have been devoted to this subject. As a generalization of local tournaments, Bang-Jensen, Huang, and Prisner [3] studied the family of in-tournaments. But in-tournaments have, as yet, received little attention. Except for the above mentioned article of BangJensen, Huang, Prisner [3], these digraphs have only been investigated by Tewes [7], [8], [9], and Tewes, Volkmann [10], [11]. It is the purpose of this paper to give more information about the properties of in-tournaments. Very recently, we have proved [12] that every arc of a strongly connected tournament of order n (even every arc of a strongly connected n-partite tournament) is contained in a directed path of order r (n + 3) /21. The following example shows that this is no longer valid for strongly connected in-tournaments. Example 1.1 Let D consist of the cycle XIX2" . XnXl together with the arcs XIXi for 3 ~ i ~ n 1. Then it is straightforward to verify that D is a strongly connected in-tournament of order n, and that the longest path through the arc XIXn-l is only of order three. Definition 1.2 If the longest path through an arc uv consists of exactly m vertices, then we call uv an m-path arc. In this paper we shall see that every strongly connected in-tournament of order n 2: 4 has at most one 3-path arc. More general, we shall prove that if a strongly connected in-tournament D of order n contains a k-path arc for every 3 ~ k ~ m ::; n 1, then all other arcs of D belong to paths of order m + 1. Also strongly connected in-tournaments without a 3-path arc but containing a 4-path arc, have only one