Connectivity of local tournaments

For a local tournament D with minimum out-degree δ, minimum indegree δ− and irregularity ig(D), we give a lower bound on the connectivity of D, namely κ(D) ≥ (2 ·max{δ+, δ−}+ 1− ig(D))/3 if there exists a minimum separating set S such that D − S i is a tournament, and κ(D) ≥ (2 ·max{δ+, δ−}+ 2|δ+ − δ−|+ 1− 2ig(D))/3 otherwise. This generalizes a result on tournaments presented by C. Thomassen [J. Combin. Theory Ser. B 28 (1980), 142–163]. An example shows the sharpness of this result. 1 Terminology and introduction We consider finite digraphs without loops and multiple arcs. For any digraph D, the vertex set is denoted by V (D) and the arc set by A(D). By n = n(D) = |V (D)| we refer to the order or D. For a vertex x ∈ V (D) we denote by N(x) = N D (x) and N −(x) = N− D (x) the set of positive and negative neighbours of x in D, respectively. Furthermore, for a vertex set X ⊆ V (D) the notation N(X) refers to the vertex set (⋃ x∈X N (x) ) \X, and N−(X) accordingly. The out-degree d(v) = d+D(v) = |N+ D (v)| of a vertex v is the number of positive neighbours of v in D, and analogously, d−(v) = dD(v) = |N− D (v)| denotes the in-degree of v. Furthermore, δ = δ(D) = min{d+(v) : v ∈ V (D)} denotes the minimum outdegree of D, and δ− = δ−(D) the minimum in-degree. Also, we refer to the minimum degree δ(D) = min{δ+(D), δ−(D)}. For two vertex sets X, Y ⊆ V (D) we denote by X the vertex set V (D) \X and by (X, Y ) the set of arcs from X to Y . Also, we define [X, Y ] as [X, Y ] = |(X, Y )|. D[X] is the digraph induced by X in D, and D − X = D [X]. Let x1, . . . , xn be the vertices of D and D1, . . . , Dn disjoint digraphs, then H = D[D1, . . . , Dn] is ∗ Corresponding author. 272 Y. GUO, A. HOLTKAMP AND S. MILZ defined by V (H) = ⋃n i=1 V (Di) and A(H) = ( ⋃n i=1 A(Di)) ∪ {yiyj : yi ∈ V (Di), yj ∈ V (Dj), xixj ∈ A(D)}. If all arcs between two vertex sets X and Y are directed from X to Y , we write X → Y , or x → Y in case X = {x}, and say X dominates Y . Let D1 and D2 be two digraphs, then D1 dominates D2, iff V (D1) → V (D2) and we write D1 → D2. A digraph which has at least one arc between every pair of distinct vertices is called semicomplete digraph. Orientations of complete graphs are called tournament. If for every vertex x of a digraph D the sets N(x) and N−(x) both induce semicomplete digraphs, respectively, then D is called a locally semicomplete digraph. A local semicomplete digraph without cycles of length 2 is called local tournament. A digraph is called connected, iff its underlying graph is connected, and strongly connected or strong, iff there exists a directed path from any vertex to any other vertex. By a strong component of a digraph that is not strong, we refer to a maximal strong induced subdigraph. We call a vertex set S ⊂ V (D) a separating set, iff D − S is not strong. A minimal separating set is minimal with respect to inclusion, and a minimum separating set is one of minimal cardinality. Furthermore, we define the connectivity of D as κ(D) = min{|S| : S is a separating set of D}. The (global) irregularity ig(D) of a digraph D is defined as ig(D) = max{max{d+(x), d−(x)} − min{d+(y), d−(y)} : x, y ∈ V (D)}. In case ig(D) = 0 we call D a regular digraph. Thomassen [6] studied the connectivity of tournaments according to their irregularity. Theorem 1.1 (Thomassen [6], 1980). If T is a tournament with ig(T ) ≤ k, then κ(T ) ≥ ⌈ |V (T )| − 2k 3 ⌉ . (1) He also characterized the tournaments for which (1) holds with equality. Lichiardopol [5] presented a generalization of this result for oriented graphs, i.e. digraphs which have at most one arc between two vertices. Theorem 1.2 (Lichiardopol [5], 2008). If T is an oriented graph, then κ(T ) ≥ ⌈ 2δ(T ) + 2δ−(T ) + 2− n(T ) 3 ⌉ . (2) It is also shown that (2) implies (1) for tournaments and that for tournaments with δ = δ− Theorem 1.2 is an improvement of Theorem 1.1. In this work we will prove two lower bounds on the connectivity of two classes of local tournaments. One of them implies Lichiardopol’s bound for tournaments. Although local tournaments are oriented graphs, our bound gives a better approximation for the connectivity of local tournaments. 2 Local tournaments and locally semicomplete digraphs Every tournament is also a local tournament, and every local tournament is also a locally semicomplete digraph. The structure of these digraphs has been studied by CONNECTIVITY OF LOCAL TOURNAMENTS 273 Bang-Jensen [1] and Guo and Volkmann [4]. A collection of their results and proofs can be found in [3]. The following results will be helpful for the proof of our main result in the next section. Lemma 2.1 (Bang-Jensen [1], 1990). Let D be a strong locally semicomplete digraph and let S be a minimal separating set of D. Then D − S is connected. According to this property, it is helpful to study the structure of connected locally semicomplete digraphs that are not strong. Since local tournaments are locally semicomplete digraphs, the following results hold for local tournaments as well. Theorem 2.2 (Bang-Jensen [1], 1990). Let D be a connected locally semicomplete digraph that is not strong. Then the following holds for D. 1. If A and B are distinct strong components of D with at least one arc between them, then either A → B or B → A. 2. If A and B are strong components of D such that A → B, then A and B are semicomplete digraphs. 3. The strong components of D can be ordered in a unique way D1, D2, . . . , Dp such that there are no arcs from Dj to Di for j > i, and Di dominates Di+1 for i = 1, 2, . . . , p− 1. For a digraph D fulfilling the condition of Theorem 2.2 the unique ordering of its strong components is called the acyclic ordering of the strong components of D. Theorem 2.3 (Guo, Volkmann [4], 1994). Let D be a connected locally semicomplete digraph that is not strong and let D1, . . . , Dp be the acyclic ordering of the strong components of D. Then D can be decomposed into r ≥ 2 induced subdigraphs D′ 1, D ′ 2, . . . , D ′ r which satisfy the following properties. 1. D′ 1 = Dp and D ′ i consists of some strong components of D and is semicomplete for i ≥ 2. 2. D′ i+1 dominates the initial component of D ′ i and there exists no arc from D ′ i to D′ i+1 for i = 1, . . . , r − 1. 3. If r ≥ 3, then there is no arc between D′ i and D′ j for i, j satisfying |i− j| ≥ 2. The unique sequence D′ 1, D ′ 2, . . . , D ′ r is called the semicomplete decomposition of D. Finally, using the semicomplete decomposition defined in Theorem 2.3 the next lemma determines the structure of locally semicomplete digraphs that are not semicomplete. Lemma 2.4 (Bang-Jensen, Guo, Gutin, Volkmann [2], 1997). If a strong locally semicomplete digraph D is not semicomplete, then there exists a minimal separating set S such that D − S is not semicomplete. Furthermore, if D1, D2, . . . , Dp is the acyclic ordering of the strong components of D − S and D′ 1, D′ 2, . . . , D′ r is the semicomplete decomposition of D − S, then r ≥ 3, D[S] is semicomplete and we have Dp → S → D1. 274 Y. GUO, A. HOLTKAMP AND S. MILZ