Hamiltonian Cycles in Cayley Graphs of Gyrogroups

: In this study, we investigate Hamiltonian cycles in the right-Cayley graphs of gyrogroups. More speciﬁcally, we give a gyrogroup version of the factor group lemma and show that some right-Cayley graphs of certain gyrogroups are Hamiltonian.


Introduction
The gyrogroup structure is a group-like structure discovered by Ungar during his study of Einstein's relativistic velocity addition law, see [1,2] for more details. A gyrogroup can be considered as a generalization of a group, where the associative property is replaced by the left gyroassociative property and the left loop property. The past decade has seen a rise in research interest regarding algebraic properties of gyrogroups and topological properties of topological gyrogroups.
As for combinatorial properties, Cayley graphs of gyrogroups were first studied by Bussaban, Kawekhao, and Suantai in [3]. After that, some of us have studied some relationships between algebraic properties of gyrogroups and combinatorial properties of the Cayley graphs, see [4,5]. In these two studies, definitions of left and right Cayley graphs of gyrogroups were given, and some properties such as transitivity, connectedness, and preservation of edge coloring have been explored.
In this research, we continue our investigation on the right-Cayley graphs of gyrogroups. In particular, we study Hamiltonian cycles in the graphs. We state and prove a gyrogroup version of the factor group lemma, and we also show that the right-Cayley graphs R-Cay(G(n), {1, m}) of the gyrogroups G(n) constructed in [6] are Hamiltonian.
Outline of the paper: In Section 2, preliminary knowledge, including the definition of a gyrogroup, the definition of its right-Cayley graph, and necessary results of these two structures, are provided. In Section 3, we give the statement and the proof of the factor group lemma, then the statement and the proof of its gyrogroup version, and some examples regarding the gyrogroup version. Later in this section, we show that the Cayley graphs of the gyrogroups constructed in [6], with the generating sets also given there, are Hamiltonian. Lastly, in Section 4, we discuss the results and leave some questions.

Background
This section contains basic knowledge of gyrogroups and their Cayley graphs. The section consists of two parts. In the first part, the definition of a gyrogroup, some important algebraic identities, and related properties are included. In the second part, the definitions of right Cayley graphs and some properties that were studied in [3][4][5] are provided.

1.
There is a unique identity element e ∈ G such that e ⊕ x = x = x ⊕ e for all x ∈ G.

2.
For each x ∈ G, there exists a unique inverse element x ∈ G such that

(left loop property)
For all elements a, b, c in a gyrogroup G, the gyroautomorphism gyr[a, b] is given by the following identity: (gyrator identity) Algebraic properties of gyrogroup parallel to those of groups were rigorously studied by Suksumran and his colleagues. Among their work, the following definitions and theorems are necessary to our work. Readers are recommended to see [7] for more details. In this study, we focus on finite gyrogroups, and the following class of gyrogroups which are constructed in [6] will be used throughout. Example 1. In [6], Mahdavi, Ashrafi, Salahshour, and Ungar constructed a class of gyrogroups whose every proper subgyrogroup is either a cyclic or a dihedral group. They call the gyrogroups in this class dihedral gyrogroups because the (normal) subgyrogroup lattice of each gyrogroup in this class is isomorphic to the (normal) subgroup lattice of the dihedral group with the same order. Later in this paper, we will see a similarity between the Cayley graphs of the dihedral gyrogroups and those of the dihedral groups. We show the construction of the dihedral gyrogroups in this example. For an integer n ≥ 3, let P(n) = {0, 1, . . . , 2 n−1 − 1}, H(n) = {2 n−1 , 2 n−1 + 1, . . . , 2 n − 1}, and G(n) = P(n) ∪ H(n). Let m = 2 n−1 . The binary operation of the gyrogroup (G(n), ⊕) is defined as follows: where t, s, k ∈ P(n) are the following non-negative integers: In [6], the gyroaddition tables and the gyration tables of G(3) and G(4) are provided, and we include them here as Tables 1-3 for reference.    I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  1  I  I  I  I  I  I  I  I  A  A  A  A  A  A  A  A  2  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  3  I  I  I  I  I  I  I  I  A  A  A  A  A  A  A  A  4  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  5  I  I  I  I  I  I  I  I  A  A  A  A  A  A  A  A  6  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  7  I  I  I  I  I  I  I  I Let us turn to some algebras of gyrogroups. To solve the equation x ⊕ a = b for x, Ungar introduced a second binary operation in G called the gyrogroup coaddition or coaddition , defined by Many identities regarding the gyrogroup addition and coaddition have been studied in [1], and we list some of them here.

Theorem 3 ([1]
). Let G be a gyrogroup. For all a, b, c ∈ G, the following properties hold:

Cayley Graphs of Gyrogroups
In this subsection, we give the definition of a right-Cayley graph of a gyrogroup and collect some properties that will be used to prove our results. For more details about the study of Cayley graphs of gyrogroups so far, we encourage readers to see [3][4][5].
A directed graph is an ordered pair D = (V, E), where V is a set of vertices and E ⊆ {(u, v)|u, v ∈ V and u = v} is a set of edges. In this paper, we will often write u → v instead of (u, v) to emphasize the direction. A directed graph is said to be undirected if, for any vertices u and v, u → v implies v → u. Given a directed graph D, we may also Definition 3 (Right-Cayley graph). Let G be a gyrogroup and let S be a subset of G not containing the identity. The (color) right-Cayley graph or (color) R-Cayley graph of G generated by S, denoted by R-Cay(G, S), is a directed graph whose vertices are the gyrogroup elements, and for any two vertices u and v, there is an edge u → v with color s if v = u ⊕ s for some s ∈ S. We will conflate the gyrogroup elements and the vertices of graph whenever there are no confusions.
The left-Cayley graphs can be defined in the same way as in Definition 3 by adding s to the left-hand side instead. We give our attention to the right-Cayley graphs because of their connection to the L-subgyrogroups. For example, Theorem 4 shows a relationship between the cosets of L-subgyrogroups and the connected components of R-Cayley graphs. Unlike groups, transitivity of the Cayley graphs of gyrogroups does not always hold, due to the lack of associativity. In the case of groups, the essence is that a left (right)multiplication by an element is always an automorphism on the right (left)-Cayley graph. For gyrogroups, some of us showed in [5] that, under a certain condition, any left-addition by an element is an automorphism on the right-Cayley graph. For the last part of this subsection, we talk about normal subgyrogroups and give the definition of a Cayley graph of a quotient gyrogroup. A gyrogroup homomorphism is a map between two gyrogroups that preserves the gyrogroup operations. Let ϕ : G → K be a gyrogroup homomorphism. The kernel of ϕ is the set {a ∈ G | ϕ(a) = e K }, where e K is the identity of K. A normal subgyrogroup H of a gyrogroup G is defined to be the kernel of a gyrogroup homomorphism with domain G, denoted by H G. In this case, the quotient space of the left cosets G/H is a gyrogroup (Theorem 29 in [7]). The gyroaddition in the quotient space is defined by We delay examples of Cayley graphs of gyrogroups and Cayley graphs of quotient gyrogroups to the next section, where our results are given.

Hamiltonian Cycles in Right-Cayley Graphs of Gyrogroups
In this section, we give our results and examples. We begin with some notations in graph theory. We may refer to a walk in a graph by specifying its vertices as v 1 → v 2 → · · · → v n , or by specifying its edge labelling as [s 1 , s 2 , . . . , s n ]. However, in the case of edge labelling, a walk is not unique unless the initial vertex is given. In this case, we will write v 1 [s 1 , s 2 , . . . , s n ] to indicate that the initial vertex is v 1 . We will write [s 1 , s 2 , . . . , s n ] i for a walk that repeats edge labelling [s 1 , s 2 , . . . , s n ] i times.
We are ready to talk about our first result in this work. For groups, the factor group lemma gives a sufficient condition for a Cayley graph of a group to be Hamiltonian. We state the lemma and provide a proof that can be adapted for the gyrogroup version.
Lemma 1 (Factor Group Lemma, Section 2.2 in [10]). Let G be a finite group and let S be a generating set of G. Suppose that there exist elements s 1 , s 2 , . . . , s n in S satisfying the following two conditions: Proof. Left-multiplying each vertex in the right-Cayley graph by an element defines an automorphism on the graph. Thus, we will show that, starting with the identity element 1, the walk 1[s 1 , s 2 , . . . , s n ] |N| is a cycle containing all elements in G, e.g., a Hamiltonian cycle. Then, for any a ∈ G, a[s 1 , s 2 , . . . , s n ] |N| is a Hamiltonian cycle. By the first condition of the lemma, the last vertex of the walk 1[s 1 , s 2 , . . . , s n ] |N| is 1(s 1 s 2 · · · s n ) |N| = 1s |N| = 1 and the length of the walk is n|N| = [G : N]|N| = |G|. Each non-identity vertex u in the walk is of form s m x, where 0 ≤ m ≤ |N| − 1 and x = s 1 , s 1 s 2 , . . . , or s 1 s 2 · · · s n−1 , see Figure 1 for reference. The second condition tells us that the n N-cosets are N, s 1 N, s 1 s 2 N, . . . , and (s 1 s 2 · · · s n−1 )N. Thus, during the walk, the N-coset of the vertex that we are visiting changes to the corresponding coset. We claim that we visit different vertices until we are back to 1. If not, suppose that vertices u and v are in distinct positions in the walk and u = v. Both of them must be in the same N-coset, say u = s i x and v = s j x, where i = j, 0 ≤ i, j ≤ |N| − 1, and x = s 1 , s 1 s 2 , . . . , or s 1 s 2 · · · s n−1 . Then, we have s i−j = 1, which is contrary to the order of s. We conclude that the walk 1[s 1 , s 2 , . . . , s n ] |N| is a Hamiltonian cycle. Notice that associativity plays an important role in the proof of the lemma. It is so powerful that we can omit parentheses entirely. However, the property is absent in gyrogroups. In particular, (((x ⊕ y) ⊕ x) ⊕ y) does not always equal (x ⊕ y) ⊕ (x ⊕ y). For example, in G(3), ((4 ⊕ 5) ⊕ 4) ⊕ 5 = 0, whereas (4 ⊕ 5) ⊕ (4 ⊕ 5) = 2. Hence, (· · · ((s ⊕ s 1 ) ⊕ s 2 ) ⊕ · · · s n−1 ) ⊕ s n is not necessary equal to s ⊕ s. Moreover, in the lemma, the cycles are given by edge labelling without specifying the initial vertex. This is due to the fact that the left multiplications are automorphisms on the Cayley graphs of groups. However, this is not the case for gyrogroups, see some examples in [4]. The following theorem is a version of this lemma for gyrogroups. Theorem 6 (Factor Gyrogroup Lemma). Let G be a finite gyrogroup and let S be a subset of G. Suppose that there exist elements (possibly repeated) s 1 , s 2 , . . . , s n in S satisfying the following conditions:
the quotient right-Cayley graph R-Cay(G/H, S * ) is Hamiltonian with the cycle Then, R-Cay(G, S) is Hamiltonian with the cycle Proof. For the first part, the second condition in this theorem together with the right cancellation law I enable us to follow the same proof as in Lemma 1. For the second part, suppose gyr[g, g ](S) = S, for all g, g ∈ G. By Theorem 5, the left addition by x is an automorphism on R-Cay(G, S). Moreover, the edge connecting x ⊕ ms and (x ⊕ ms) ⊕ s 1 is gyr[x, ms]s 1 , and the edge connecting x ⊕ (· · · ((ms ⊕ s 1 ) ⊕ s 2 ) · · · ⊕ s q ) and x ⊕ ((· · · ((ms ⊕ s 1 ) ⊕ s 2 ) · · · ⊕ s q ) ⊕ s q+1 ) is gyr[x, ((ms ⊕ s 1 ) ⊕ s 2 ) · · · ⊕ s q ]s q+1 obtained by moving the outermost parenthesis to the left-hand side, where m = 0, 1, . . . , |H| − 1 and q ≤ n − 1. Adding 4 to the generating set, we have a new generating set S 2 = {1, 4, 6} that satisfies the condition gyr[g, g ](S 2 ) = S 2 for all g, g ∈ G(3). Theorem 6 implies that, for any x ∈ G(3), the cycle starting at x, which is 1 [1,6,1,4,1,6,1,4], is Hamiltonian. The right Cayley graph R-Cay(G(3), S 2 ) is depicted in Figure 3.  Proof. We will show that the walk exhausts all vertices in H(n) then all vertices in P(n), see Figure 4 for reference. The walk starts at the vertex 0 and, after adding to the right by m = 2 n−1 , we are at the vertex m ∈ H(n). All vertices in the walk starting at m, m[1] m−1 , are in H(n) since h ⊕ 1 ∈ H(n) whenever h ∈ H(n) by the definition of addition in G(n). We observe that h ⊕ 1 = s + m, where s ≡ h + 2 n−2 − 1 mod m and 0 ≤ s ≤ m − 1, and also observe that gcd(m, 2 n−2 − 1) = gcd(2 n−1 , 2 n−2 − 1) = 1. Hence, all vertices in the walk m[1] m−1 are all distinct and all of H(n). The terminal vertex of the path m[1] m−1 is s + m, where s ≡ (2 n−2 − 1)(2 n−1 − 1) mod m. Adding m to the right of the terminal vertices of the path m[1] m−1 , we obtain (s + m) ⊕ m = 1 ∈ P(n) because It is easy to see that 1 [1] m−1 is the path 1 → 2 → 3 → · · · → m − 1 → 0 consisting of all elements in P(n).

Discussion
With the absence of associativity, many properties of Cayley graphs of groups do not hold true for gyrogroups. However, similar to other algebraic and topological properties of gyrogroups that have been studied, when imposed with the so-called strongly generated property (S is a generating set of a gyrogroup G and gyr[g, g ](S) = S for all g, g ∈ G), many group-like properties of gyrogroups are valid to some degree. In our version of the factor gyrogroup lemma, when the strongly generated property holds, the Hamiltonian cycle is independent of the choice of the starting vertex with a specific change in the edge labelling. While many properties of Cayley graphs of groups have been studied, their gyrogroup counterparts are yet to be explored. We would like to end our discussion with some questions similar to the classical ones in group theory and a question of our interest.

Question 1. Is every connected right-Cayley graph of a gyrogroup Hamiltonian?
In group theory, Question 1 is still an open problem. The progress of the study of Hamiltonicity in Cayley graphs of groups can be found in [11][12][13][14], for instance. Therefore, we also ask a more specific question.

Question 2. Is every connected right-Cayley graph of a dihedral gyrogroup Hamiltonian?
We remark that, even in the case of groups, Question 2 is not completely solved. In [15], the authors prove that the right-Cayley graph of a dihedral group D 2n is Hamiltonian for all even integers n (see Corollary 4.1 of [15]). Seeing the similarity between the Cayley graphs of dihedral gyrogroups and the Cayley graphs of dihedral groups, we are interested in the following question: