An optimal lower bound for the size of periodic digraphs

A periodic digraph is the digraph associated with a periodic point of a continuous map from the unit interval to itself. This digraph encodes “covering” relation between minimal intervals in the corresponding orbit, which allows the application of purely combinatorial arguments in establishing results on the existence and co-existence of periods of periodic points (for example, in proving the famous Sharkovsky’s theorem). In this article, an optimal lower bound for the size of periodic digraphs is provided and thus some previous results of Pavlenko are tightened.


Introduction
Combinatorial dynamics is a branch of dynamical systems that studies the existence of periodic points and the structure of their orbits for self-maps on various structures. A celebrated result that helped to shape this field is the Sharkovsky's theorem [8] which completely describes the coexistence of periods of periodic points of a continuous map from the unit interval to itself. It turned out that one can prove Sharkovsky's theorem using purely combinatorial arguments [2,9]. These ideas are based on the following construction. Let f : [0, 1] → [0, 1] be a continuous map and x ∈ [0, 1] be its n-periodic point. Clearly, the restriction of f to the corresponding orbit orb f (x) = {x, f (x), . . . , f n−1 (x)} is a cyclic permutation of the latter. Consider the natural ordering orb f (x) = {x 1 < · · · < x n }. The corresponding periodic digraph Γ has the vertex set The idea behind this construction is the following, each vertex i ∈ V (Γ) corresponds to a minimal interval [x i , x i+1 ], and there is an arc i → j in Γ provided the interval [x i , x i+1 ] "covers" interval [x j , x j+1 ] under f . This construction can be extended to a continuous vertex maps on topological trees, which enables obtain Sharkovsky-type results (not linear, but partial orders) in this more general setting [1].
The purely graph-theoretic properties of periodic digraphs were studied by Pavlenko in his three papers [5][6][7]. For example, the number of non-isomorphic periodic graphs with a given number of vertices was obtained in [5]. Characterizations of periodic digraphs and their induced subgraphs were proved in [6] and [7], respectively.

Preliminaries
A graph is a pair G = (V, E), where V = V (G) is the set of its vertices and E = E(G) is the set of its edges which are unordered pairs of vertices. For convenience, instead of {u, v} we will write uv for an edge in a graph. For a set of vertices we denote the set of edges of G whose vertices are in V .
A graph is connected provided there is a path between every pair of its vertices. The vertex set V (G) of a connected graph G is naturally equipped with the standard metric d G , where d G (u, v) equals the length of a shortest path u−v path in G. For a pair of vertices u, v ∈ V (G) in a connected graph G, the metric interval between u, v is the set The Wiener index of a connected graph is the value A tree is a connected graph without cycles. Prominent example of trees are path graphs P n , where V (P n ) = {1, . . . , n} and E(P n ) = {ij : | are called the out-degree and the in-degree of u, respectively. Now let X be a tree and σ : V (X) → V (X) be a map from V (X) to itself. The corresponding Markov graph is a digraph Γ = Γ(X, σ) with the vertex set V (Γ) = E(X) and the arc set The Markov graph Γ(X, σ) is depicted in Figure 1.
Denote by T (X), P(X) and C(X) the classes of all maps, permutations and cyclic permutations of V (X), respectively. The average number size of Markov graphs for maps in these classes can be explicitly calculated in terms of the Wiener index of X as follows.
Theorem 2.1. [4] For any tree X with n ≥ 3 vertices the next equalities hold: An n-periodic digraph is a Markov graph Γ(X, σ) for a path X = P n and its cyclic permutation σ. The number of non-isomorphic n-periodic digraphs was obtained in [5].
Also, periodic digraphs admit a nice graph-theoretic characterization. To present this result, for any digraph D, we define a self-map on the power set A : 2 V (Γ) → 2 V (Γ) in such a way: . For a collection of sets F, the corresponding intersection graph is an undirected graph with the vertex set F and two sets A, B ∈ F are being adjacent provided A ∩ B = ∅. Theorem 2.3. [6] A digraph Γ with n ≥ 1 vertices is a periodic digraph if and only if there exists a vertex u ∈ V (Γ) with d − Γ (u) = 2 such that the singleton {u} is an (n + 1)-periodic point for A, and the intersection graph for the collection of sets orb A ({u}) is a path.
The following bounds on the size of n-periodic digraphs can be easily obtained by examining in-degrees of their vertices. Proposition 2.1. [6] Let Γ be an n-periodic digraph with n ≥ 3. Then n ≤ |A(Γ)| ≤ n 2

.
However, checking all the 5-periodic digraphs (a complete list of these digraphs can be found in [9]), we can conclude that the bounds from Proposition 2.1 are not optimal (every 5-periodic digraph have at least 6 arcs and at most 11 arcs). The aim of this paper is to present an optimal lower bound for the size of n-periodic digraphs.

Main result
The main result of this note is the following theorem.
Hence, in all cases we have d − Γ (e k ) ≥ 2. In other words, i ∈ V 1 and i = n 2 implies n − i / ∈ V 1 . This clearly means that |V 1 | ≤ n 2 . Therefore, To prove the optimality of a given bound, let us construct a cyclic permutation which realizes this bound. At first, let n be an even number. If n = 2, then the unique cyclic permutation of V (X) produces a periodic digraph with exactly 3n−3 2 = 1 arc. Now assume n ≥ 4. Consider the map It is easily seen that σ is a permutation of V (X). To prove that σ is a cyclic permutation, we show that for any Similarly, if n 2 + 1 ≤ k ≤ n − 1, then m = 2k − n + 1. Finally, σ(1) = n. Putting Γ = Γ(X, σ), we obtain This asserts the equality Now let n be an odd number. In this case, consider the map n + 1 2 , if i = n for 1 ≤ i ≤ n. One can similarly show that σ is a cyclic permutation of V (X). We have   Figure 2), the corresponding periodic digraph Γ(P 7 , σ) has 3·7−3 2 = 9 arcs (see Figure 3).