Distinguishing number and distinguishing index of Kronecker product of two graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The Kronecker product $G\times H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(u, x), (v, y)\} | \{u, v\} \in E(G) ~and ~\{x, y\} \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of Kronecker product of two graphs.


Introduction and definitions
Let G = (V, E) be a simple graph of order n 2. We use the the following notations: The set of vertices adjacent in G to a vertex of a vertex subset W ⊆ V is the open neighborhood N G (W ) of W . Aut(G) denotes the automorphism group of G. A labeling of G, φ : V → {1, 2, . . . , r}, is said to be r-distinguishing, if no non-trivial automorphism of G preserves all of the vertex labels. The point of the labels on the vertices is to destroy the symmetries of the graph, that is, to make the automorphism group of the labeled graph trivial. Formally, φ is r-distinguishing if for every non-trivial σ ∈ Aut(G), there exists x in V such that φ(x) = φ(xσ). The distinguishing number of a graph G is defined by D(G) = min{r| G has a labeling that is r-distinguishing}. This number has defined by Albertson and Collins [1]. Similar to this definition, Kalinowski and Pilśniak [15] have defined the distinguishing index D ′ (G) of G which is the least integer d such that G has an edge colouring with d colours that is preserved only by a trivial automorphism. If a graph has no nontrivial automorphisms, its distinguishing number is 1. In other words, D(G) = 1 for the asymmetric graphs. The other extreme, D(G) = |V (G)|, occurs if and only if G = K n . The distinguishing index of some examples of graphs was exhibited in [15]. For instance, D(P n ) = D ′ (P n ) = 2 for every n 3, and D(C n ) = D ′ (C n ) = 3 for n = 3, 4, 5, D(C n ) = D ′ (C n ) = 2 for n 6. It is easy to see that the value |D(G) − D ′ (G)| can be large. For example D ′ (K p,p ) = 2 and D(K p,p ) = p + 1, for p ≥ 4. A graph and its complement, always have the same automorphism group while their graph structure usually differs, hence D(G) = D(G) for every simple graph G. The distinguishing number and the distinguishing index of some graph products has been studied in literature (see [2,3,4,11,12]). The Cartesian product of graphs G and H is a graph, denoted G✷H, whose vertex set is V (G) × V (H). Two vertices (g, h) and (g ′ , h ′ ) are adjacent if either g = g ′ and hh ′ ∈ E(H), or gg ′ ∈ E(G) and h = h ′ . Denote G✷G by G 2 , and recursively define the k-th Cartesian power of G as G k = G✷G k−1 . A non-trivial graph G is called prime if G = G 1 ✷G 2 implies that either G 1 or G 2 is K 1 . Two graphs G and H are called relatively prime if K 1 is the only common factor of G and H. We need knowledge of the structure of the automorphism group of the Cartesian product, which was determined by Imrich [10], and independently by Miller [17].
The Kronecker product is one of the (four) most important graph products and seems to have been first introduced by K.Čulik, who called it the cardinal product [8]. Weichsel [20] proved that the Kronecker product of two nontrivial graphs is connected if and only if both factors are connected and at least one of them possesses an odd cycle. If both factors are connected and bipartite, then their Kronecker product consists of two connected components. The Kronecker product G × H of graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(u, x), (v, y)}|{u, v} ∈ E(G) and {x, y} ∈ E(H)}. The terminology is justified by the fact that the adjacency matrix of a Kronecker graph product is given by the Kronecker matrix product of the adjacency matrices of the factor graphs; see [20] for details. However, this product is also known under several different names including categorical product, tensor product, direct product, weak direct product, cardinal product and graph conjunction. The Kronecker product is commutative and associative in an obvious way. It is computed that |V (G × H)| = |V (G)|.|V (H)| and |E(G × H)| = 2|E(G)|.|E(H)|. We recall that graphs with no pairs of vertices with the same open neighborhoods are called R-thin. In continue, we need the following theorem: Suppose ϕ is an automorphism of a connected non-bipartite R-thin graph G that has a prime factorization G = G 1 × G 2 × . . . × G k . Then there exists a permutation π of {1, 2, . . . , k}, together with isomorphisms ϕ i : G π(i) → G i , such that ϕ(x 1 , x 2 , . . . , x k ) = (ϕ 1 (x π(1) ), ϕ 2 (x π(2) ), . . . , ϕ k (x π(k) )).

Distinguishing number of Kronecker product of two graphs
We begin with the distinguishing number of Kronecker product of complete graphs.
Theorem 2.1 Let k, n, and d be integers so that d 2 and (d − 1) k < n d k . Then Proof. It is easy to see that K k × K n is the complement of Cartesian product K k ✷K n .
It is known that connected non-bipartite graphs have unique prime factor decomposition with respect to the Kronecker product [16]. If such a graph G has no pairs u and v of vertices with the same open neighborhoods, then the structure of automorphism group of G depends on that of its prime factors exactly as in the case of the Cartesian product. As said before graphs with no pairs of vertices with the same open neighborhoods are called R-thin and it can be shown that a Kronecker product is R-thin if and only if each factor is R-thin.
Theorem 2.2 Let G and H be two simple connected, relatively prime graphs, nonbipartite R-thin graphs, then D(G × H) = D(G✷H).
Proof. By hypotheses and Theorems 1.1 and 1.2, it can be concluded that Aut(G × H) = Aut(G✷H). Therefore D(G × H) = D(G✷H).
Imrich and Klavzar in [12] proved that the distinguishing number of k-th power with respect to the Kronecker product of a non-bipartite, connected, R-thin graph different from K 3 is two.

Theorem 2.3 [12]
Let G be a nonbipartite, connected, R-thin graph different from K 3 and ×G k the k-th power of G with respect to the Kronecker product.
Now we want to obtain the distinguishing number of Kronecker product of two complete bipartite graphs. We need the following lemma: Proposition 2.5 If K m,n and K p,q are complete bipartite graphs such that q p and m n then the distinguishing number of K m,n × K p,q is D(K m,n × K p,q ) = mq + 1 m = n, p = q mq otherwise.
Proof. The Kronecker product K m,n × K p,q is disjoint union of two complete bipartite graphs K mp,nq and K mq,np by Lemma 2.4. Hence if m = n and p = q, then K mp,nq and K mq,np are the two non-isomorphic graphs, and so D(K m,n × K p,q ) = max{D(K mp,nq ), D(K mq,np )} = mq. If m = n or p = q, then K mp,nq and K mq,np are isomorphic to K mp,mq or K mp,np , respectively. In fact Thus using the value of the distinguishing number of complete bipartite graphs we have D(K m,n × K p,q ) = mq m = n, p = q or m = n, p = q mq + 1 m = n, p = q.
Therefore the result follows.
Corollary 2.6 Let m, n 3 be two integers. The distinguishing number of Kronecker product of star graphs K 1,n and K 1,m is D(K 1,n × K 1,m ) = mn.
The following result shows that the distinguishing number of Kronecker product of complete bipartite graphs is an upper bounds for the distinguishing number of Kronecker product of bipartite graphs.
Proof. It is sufficient to note that Aut(G × H) ⊆ Aut(K m,n × K p,q ), and G × H and K m,n × K p,q have the same size. Now we have the result by Proposition 2.5.
Before we prove the next result we need some additional information about the distinguishing number of complete multipartite graphs. Let K a 1 j 1 ,a 2 j 2 ,...,ar jr denote the complete multipartite graph that has j i partite sets of size a i for i = 1, 2, . . . , r and a 1 > a 2 > . . . > a r .
Theorem 2.9 If G and H are two simple connected, relatively prime graphs such that G × H has j i R-equivalence classes of sie a i for i = 1, . . . , r, and a 1 > a 2 > . . . > a r then Proof. Since Aut(G✷H) ⊆ Aut(G×H), so D(G✷H) ⊆ D(G×H). To prove the second inequality, it is sufficient to consider each R-equivalence classes of G × H as a partite set. Thus graph G × H can be considered as multipartite graph that has j i partite sets of size a i such that every two partite sets of this multipartite graph is complete bipartite or there exists no edge between the two partite sets. So the automorphism group of this multipartite graph is subset of the automorphism group of complete multipartite graph with the same partite sets. Therefore D(G × H) D(K a j 1 1 ,...,a jr r ), and the result follows by Theorem 2.8.
By using the concept of the Cartesian skeleton we can obtain an upper bound for Kronecker product of R-thin graphs. For this purpose we need the following preliminaries frome [9]: The Boolean square of a graph G is the graph G s with V (G s ) = V (G) and E(G s ) = {xy | N G (x) ∩ N G (y) = ∅}. An edge xy of the Boolean square G s is dispensable if it is a loop, or if there exists some z ∈ V (G) for which both of the following statements hold: The Cartesian skeleton S(G) of a graph G is the spanning subgraph of the Boolean square G s obtained by removing all dispensable edges from G s .

Distinguishing index of Kronecker product of two graphs
In this section we investigate the distinguishing index of Kronecker product of two graphs. Let us start with the Kronecker product power of K 2 . It can be seen that ×K n 2 is disjoint union of 2 n−1 number of K 2 , and so D ′ (×K n 2 ) = 2 n−1 for n 2. Let ij be the notation of the vertices the Kronecker product P m × P n of two paths of order m and n where 0 i m − 1 and 0 j n − 1. From [14], we know that P m × P n is bipartite, the number of vertices in even component of P m × P n is ⌈mn/2⌉ while that in odd component is ⌊mn/2⌋ (see Figure 1). It can be easily computed that distinguishing index of P m × P n is two, unless D ′ (P 3 × P 2 ) = 3 and D ′ (P 3 × P 3 ) = 4, because P 3 × P 2 is disjoint union P 3 ∪ P 3 , and P 3 × P 3 is disjoint union K 1,4 ∪ C 4 .
The distinguishing index of the square of cycles and for arbitrary power of complete graphs with respect to the Cartesian, Kronecker and strong product has been considered by Pilśniak [18]. In particular, she proved that D ′ (×C 2 m ) = 2 for the odd value of m 5, and D ′ (×K r n ) = 2 for any n 3 and r 2. Let us state and prove the following lemma concerning the Kronecker product K 2 × H.
Proof. Let f be an automorphism of bipartite graph K 2 ×H with partite sets {(v 1 , x)|x ∈ V (H)} and {(v 2 , x)|x ∈ V (H)}. Since f preserves the adjacency and non-adjacency re- Let L be a distinguishing edge labeling of H. If (v 1 , h 1 )(v 2 , h 2 ) be an arbitrary edge of K 2 × H, then we assign it the label of the edge h 1 h 2 in H. Now suppose that hh ′ is an edge of H and fix it. We change the label of the edge (v 1 , h)(v 2 , h ′ ) to a new label.
If f is an automorphism of K 2 × H preserving the labeling, then with respect to the label of the two edges (v 1 , h)(v 2 , h ′ ) and (v 1 , h ′ )(v 2 , h) we have f (v i , h) = (v i , ϕ(x)) for i = 1, 2 and some ϕ ∈ Aut(H). On the other hand ϕ is the identity, because we labeled the edges of K 2 × H by the distinguishing edge labeling L of H. Therefore this labeling is distinguishing. If H is bipartite then K 2 × H = H ∪ H, and so the result follows.
Remark 3.2 Let (G, φ) denote the labeled version of G under the labeling φ. Given two distinguishing k-labelings φ and φ ′ of G, we say that φ and φ ′ are equivalent if there is some automorphism of G that maps (G, φ) to (G, φ ′ ). We denote by D(G, k) the number of inequivalent k-distinguishing labelings of G which was first considered by Arvind and Devanur [5] and Cheng [6] to determine the distinguishing numbers of trees. In  Proof. The consecutive vertices of P m are denoted by 0, 1, . . . , m − 1 and the central vertex of K 1,n by 0, and its pendant vertices by 1, . . . , n. Since K 1,n and P m are connected and bipartite, their Kronecker product consists of two connected components (see Figure 2). If m is odd, then we label the edges (10, 0i) and (j0, (j + 1)i) with label i where 1 i n and 0 j m − 1. We label the remaining edges with an arbitrary label, say 1. Using Figure 2 and regarding to the number of pendant vertices we can obtain that this labeling is distinguishing and D ′ (K 1,n × P m ) = n. If m is even, then we label the edges (10, 0i), (00, 1i), and (j0, (j + 1)i) with label i for 1 i n and 2 j m − 2. Also we label the edges ((m − 2)0, (m − 3)i) with label 1 and the edges (10, 2i) with label 2 for 1 i n. We label the remaining edges with an arbitrary label, say {z ′ 1 , . . . , z ′ D ′ (H) }, then we have a distinguishing edge labeling of X by Theorem 1.2, and so the result follows.
Theorem 3.6 Let X be a connected non-bipartite R-thin graph which has a prime factorization X = G × H where G and H are simple and D ′ (G) = D ′ (H) = 1. Then D ′ (X) 2.
Proof. If G and H are non-isomorphic then |Aut(X)| = 1 by Theorem 1.2, and so D ′ (X) = 1. Otherwise, D ′ (G × G) = 2 because, G × G is a symmetric graph and we have a 2-distinguishing edge labeling of it as follows: according to notations of Theorem 3.5, if we label the all elements of E 11 1 with label 1, and all elements of E 1j 1 with label 2 for every 2 j |E(G)|, then we have a 2-distinguishing edge labeling of G × G.
Corollary 3.7 If G is a connected non-bipartite R-thin graph with D ′ (G) = 1, then D ′ (×G k ) = 2 for any k 2.
Proof. It follows immediately by induction on k and Theorems 3.5 and 3.6.