The non-commuting graph of a non-central hypergroup

Abstract The aim of this paper is to construct and study the properties of a certain graph associated with a non-central hypergroup, i.e. a hypergroup having non-commutative the associated fundamental group. The method of the construction of the graph is similar to that one proposed by Paul Erdős, when he defined a graph associated with a non-commutative group. We establish necessary and /or sufficient conditions for the associated graph to be Hamiltonian or planar.


Introduction
It is much simpler to deal with a concrete problem, to nd its solution if we can somehow graphically represent it, or at least part of it. As written in the book written by Bondy and Murty [1], "many real-world situations can conveniently be described by means of a diagram consisting of a set of points together with lines joining certain pairs of these points". This is the intuitive de nition of the concept of graph, born in 1736 when Euler was asked to nd a nice path across the seven Köningsberg bridges (for more details see Sudakov [2]). An Eulerian path crosses over each of the seven bridges exactly once. Since then graph theory has established itself as a fundamental branch of mathematics, as an important tool for solving theoretical and practical problems in combinatorics, operational research, chemistry, genetics, geography, architecture and engineering, etc. In recent years, several connections in both directions between graphs and algebraic hyperstructures (in particular hypergroupoids, hypergroups or hypergraphs) have been established and developed. On one side, paths in graphs de ne di erent hyperoperations (i.e. functions that associate with any pair of elements of a nonempty set a subset of the support set) on the set of their vertices, leading to the association of algebraic hyperstructures with graphs. This direction was considered, for example, by Massouros [3], Kalampakas and Spartalis [4][5][6], Rosenberg [7], Golmohamadian and Zahedi [8]. The aim of these works is to determine necessary and/or su cient conditions such that the associated hyperstructure satis es determined properties such as associativity, reproducibility, transposition, commutativity, etc. Another problem discussed by these articles concerns the computations of the number of the associated hyperstructures.
On the other side, several studies have been conducted in the other direction of the connection between graphs and hyperstructures. This time the starting object is a hypergroupoid (a hypergroup or a hypergraph) and the result is a graph. In a recently published article (Hamidi et al. [9]), the authors de ne and compute the number of graphs obtained from hypergraphs. In this note we propose a construction of a graph starting from a non-central hypergroup, i.e. a nite hypergroup having the associated fundamental group noncommutative. A similar method was considered in 1975 by Paul Erdős (as mentioned in Neumann [10]), who de ned a graph in the following way. Let G be a non-abelian group and take the set of vertices V = G \ Z(G), where Z(G) = {x ∈ G | xy = yx, ∀y ∈ G} is the set of the elements of G commuting with all elements in G, i.e. the center of G. Then two vertices x and y are joined whenever xy ≠ yx. Extending this construction to the hypergroups framework, instead of a non-abelian group, we will consider a non-central hypergroup, i.e. the associated fundamental group H/β is non-abelian. In other words, the set of the vertices is is the identity of the group H/β} is the core (heart) of the hypergroup (H, •). We establish necessary and/or su cient conditions for the associated graph to be Hamiltonian or planar.

Preliminaries
We recall here some basic notions of graph theory (connected with Hamiltonian and planar graphs) and hypergroup theory, and we x the notations used in this note. For the rst theory we referee the readers to the fundamental book by Bondy and Murty [1] (from which we stated all the notions and results in the rst subsection), while surveys of the theory of hyperstructures can be found in the books written by Corsini [11], Davvaz and Leoreanu-Fotea [12], Corsini and Leoreanu [13] and Vougiouklis [14].

. Hamiltonian and planar graphs
A graph G is a pair G = (V , E), where V is a set of vertices and E is a (multi)set of unordered pairs of vertices, called edges. We write V(G) for the set of vertices and E(G) for the set of edges of a graph G. A loop is an edge (v, v) for some v ∈ V, so an edge that connects a vertex v to itself. An edge e = (u, v) is a multiple edge if it appears multiple times in E. A graph is simple if it has no loops or multiple edges. If e = (u, v) is an edge of a graph G, then we say that u and v are adjacent in G and that e joins u and v.
In this paper we will only consider nite simple graphs.
De nition 2. 1. Given a graph G = (V , E) and a vertex v ∈ V, we de ne the degree deg(v) of v to be the number of all its adjacent vertices.
The minimum degree of G is denoted by δ(G).

De nition 2.7.
A circuit in a graph G is a path that begins and ends at the same vertex, i.e. a closed path. A Hamiltonian circuit is a closed path that visits every vertex in the graph exactly once. A graph is Hamiltonian if it has a Hamiltonian circuit.
De nition 2. 8. Let G = (V , E) be a graph and x = v , v , . . . , vn = y be a path between two vertices x and y in G. We mean by d(x, y) the minimum length of all paths from x to y. If there are no walks between x and y, let d(x, y) = ∞ by convention. In the following we recall two types of graphs, that we will use in the next section. A simple graph that contains every possible edge between all the vertices is called a complete graph. A complete graph with n vertices is denoted as Kn.
A graph G = (V , E) is bipartite if its vertex set V can be partitioned into two sets X and Y in such a way that every edge of G has one end vertex in X and the other one in Y. In this case, X and Y are called the partite sets. A bipartite graph with partite sets X and Y is called a complete bipartite graph if the graph contains exactly all edges that have one end vertex in X and the other end vertex in Y. If there are n vertices in X and m vertices in Y, we denote it as Kn,m. Usually Kn,m and Km,n are considered to be the same.
It is obvious that each Kn is a Hamiltonian graph whenever n ≥ , while Kn,m is a Hamiltonian graph if and only if n = m ≥ .
The last part of the short overview on graphs is dedicated to the planar graphs, i.e. those graphs isomorphic with a plane graph, that is a graph drawn on the plane without edge crossing. For example, K and K , are not planar graphs, while K is a planar graph. Theorem 2. 10. For a simple connected planar graph with nv ≥ vertices and ne edges there is ne ≤ nv − .
As a corollary, we get the following result. Proposition 2. 11. A simple connected planar graph with nv ≥ vertices has a vertex of degree ve or less, i.e. δ(G) ≤ .

De nition 2.12. A planar graph
Such a graph is also called triangulated since all the faces are triangles. Every planar graph is a subgraph of a maximal planar graph. Theorem 2. 13. If G is a maximal planar graph with nv vertices and ne edges, then ne = nv − . Theorem 2.14. Let G be a maximal planar graph with nv ≥ vertices and diameter k = diam(G). Let n i denote the number of vertices of degree i in G, for i = , , . . . , k. Then n + n + n = + n + n + . . .
One graph is homeomorphic to another one if we can turn one into the other by adding or removing degree-two vertices.

Theorem 2.15. (Kuratowski , s theorem). A graph is non-planar if and only if it contains a subgraph homeomorphic to K or K , .
Planar graphs have many applications in real-world problems, for example the color theorem states that it is possible to colour the faces of a planar graph with four or fewer colours so that no two adjacent faces are colored alike.

. Hypergroups
Let us start this subsection with the de nition of a hypergroup. It is a non-empty set H endowed with a hyperoperation • : H × H −→ P * (H), satisfying the associative property, i.e. for any x, y, z ∈ H, there is , and the reproduction axiom, i.e. for any One of the key element in the hypergroup theory is the concept of heart of a hypergroup, that we will brie y recall in this subsection. It has, somehow, a similar role as the center of a group, since it commutes with any non-empty subset of the hypergroup. For a particular type of hypergroups, i.e. the complete hypergroups, the heart is the set of all bilateral identities of the hypergroup, i.e. the set of the elements e ∈ H satisfying the property x ∈ x•e∩e•x for any x ∈ H. In order to de ne the heart of a hypergroup, we need to introduce an equivalence relation, called also fundamental relation because its properties. This is the β relation. More details regarding its meaning and applications can be found, e.g. in Antampou s et al. [15], Al Tahan et al. [16], Novák et al. [17], Hamidi [18].
De ne rst, for all n ≥ , on a hypergroup (H, •) the relation βn as follows: Denote by β * the transitive closure of β, so β * is an equivalence relation on H, see Corsini [11]. It is well known that β * is the smallest strongly regular relation on a hypergroup (H, •), such that the quotient H/β * is a group with respect to the following operation H/β * is called the fundamental group associated with H. The heart ω H of the hypergroup H is the set of all elements x of H, for which the equivalence class β * (x) is the neutral element of the fundamental group H/β * . Moreover, in a hypergroup H we have for all x ∈ H. Generalyzing, for any non-empty subset B of a hypergroup H, there is In other words, considering the canonical projection φ H :

Non-commuting graph of a non-central nite hypergroup
This section is dedicated to the construction and study of the properties of a certain graph associated with a non-central hypergroup, i.e. a hypergroup having non-commutative the associated fundamental group.
In particular we search for conditions under which this graph is Hamiltonian or planar. The method of the construction of the graph is similar to that one proposed by Paul Erdős, when he de ned a graph associated with a non-commutative group G: the set of vertices is V = G \ Z(G), where Z(G) = {x ∈ G | xy = yx, ∀y ∈ G} and two vertices x and y are joined whenever xy ≠ yx. This graph was called later on by Abdollahi et al. [19] the non-commuting graph of the group G.
First we will characterise all hypergroups having non-commutative fundamental group.

De nition 3.1. Let (H, •) be a nite hypergroup and set
If T H ≠ ∅, then H is called a non-central hypergroup.
Based on the properties of the heart of a hypergroup, we immediately obtain the following characterisation of a non-central hypergroup. is non-abelian. Proof. We know that, for any x ∈ H, we have β(x) = x • ω H . From here it results the equivalence that concludes the proof.
Since any non-abelian group contains more than elements, it follows that, if a hypergroup H is non-central, i.e. the associated fundamental group H/β * in non-abelian, then its cardinality is at least .

De nition 3.3.
Let (H, •) be a non-central hypergroup. We associate with H a graph G H as follows: T H is the set of vertices and join two distinct vertices x, y whenever y The most simple case of non-central hypergroup is a non-abelian group, where the hyperoperation on the hypergroup coincides with the operation of the group. In the following we will see some examples of the graphs associated with particular non-abelian groups, in the sense of De nition 3. 3.  = {e, a, a , a , a , b, ab, a b, a b

Therefore G D has a subgraph isomorphic to K , meaning that G D is not planar, by Kuratowski's Theorem.
Notice that all previous examples are for groups, while the following one is for a proper hypergroup. Let us x now a notation. For any element x in a hypergroup (H, •) denote by x = β(x) the equivalence class of x modulo the relation β.

Proposition 3.8. Let (H, •) be a non-central hypergroup. i) If x ∈ T H , then x • ω H ⊆ T H . ii) T H
ii) It follows immediately from part i) and equivalence x ∈ T H ⇐⇒ x ∈ T H/β .
Thus |H/β| < , that is H/β is an abelian group, which is a contradiction because H is a non-central hypergroup. Besides, since |T H | ≥ , there always exists an edge between two vertices of G H , so the graph G H is connected.
The following result gives a su cient condition to have an edge between two vertices.

Lemma 3.10. If (H, •) is a non-central hypergroup and x • y • ω H ≠ y • x • ω H , for some x, y ∈ T H , then, for any a ∈ x • ω H and any b ∈ y • ω H , there exists an edge between a and b.
Proof Based on this and on the de nition of a bipartite complete graph, we get the following result.

Theorem 3.11. If H is a non-central hypergroup and (x, y) ∈ E(G H ), then Kx,y
Proof. It follows immediately from Lemma 3.10, since Kx,y can be partitioned into the sets X = x • ω H and Y = y • ω H and each edge (a, b) has one end vertex in X and the other one in Y.

Lemma 3.12. Let H be a non-central hypergroup. Then the associated graph G H is not complete.
Proof. Suppose that H is a non-central hypergroup. Let us assume by contradiction that G H is a complete graph, then G H/β is a complete graph, too. Thus diam(G H/β ) = . Now let x ∈ T H/β . Then which is false. Moreover, if y ∉ T H/β , then x ⊗ y ∈ T H/β because, otherwise, we would get that x ⊗ y commutes with every element in H/β, i.e. x ⊗ y ⊗ z = z ⊗ x ⊗ y, for all z ∈ H/β. Since y commutes with all elements of H/β, it results that x ⊗ z ⊗ y = z ⊗ x ⊗ y, so x ⊗ z = z ⊗ x, for all z ∈ H/β, which contradicts the fact that x ∈ T H/β . Thus x ⊗ y ∈ T H/β and so (x ⊗ y) − = x ⊗ y, equivalently with y − ⊗ x − = x ⊗ y = y ⊗ x (since y ∈ T H/β ), thereby y = y − . We conclude that x = x − , for all x ∈ H/β, meaning that H/β is an abelian group, which is a contradiction.

Theorem 3.13. If H is a non-central hypergroup, then diam(G H ) = .
Proof. Using Lemma 3.12, we know that the associated graph G H is not complete, so there exist x, y ∈ T H such that d(x, y) ≠ . Therefore We must consider the following two cases. 1.
Morever, using Lemma 3.10, we have Hence, by Dirac's theorem, it follows that G H is a Hamiltonian graph.

Corollary 3.16. Every non-abelian group is a Hamiltonian hypergroup.
Example 3. 17 Thus deg(v) ≥ for all v ∈ T H . According with Proposition 2.11, the graph G H is not planar since it does not consist a vertex of degree less than or equal to .
We conclude the study with two interesting properties of non-central hypergroups having a planar associated graph.

Proposition 3.22.
Let H be a non-central hypergroup. If the associated graph G H is planar, then the quotient group H/β is a non-abelian group of order less than or equal to , and Kx,y K , , for all (x, y) ∈ E(G H ).

Proof.
Suppose that H is a non-central hypergroup such that G H is a planar graph. According with Proposition 2.11, there exists x ∈ T H such that deg(x) ≤ . Therefore in the graph G H/β , we have deg(x) ≤ . Consequently |H/β \ C(x)| ≤ , where C(x) is the set of elements of H/β commuting with x. Because |C(x)| ≤ |H/β| , it results that |H/β| ≤ . If |H/β| = then H/β must be the dihedral group of order , but its associated graph is not planar (as shown in Example 3.6). Hence |H/β| ≤ (notice that every group of order is abelian). Moreover, according with Kuratowski's theorem, we have Kx,y K , , for all (x, y) ∈ E(G H ).

Theorem 3.23. Let H be a non-central hypergroup such that the associated graph G H is planar. Then H H/β.
Proof. Consider H be a non-central hypergroup such that G H is a planar graph. According with Proposition 3.22, the quotient group H/β is of order or , thus G H/β G D or G H/β G D . We consider the two following cases. 1