Connectivity and other invariants of generalized products of graphs

Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D)\longrightarrow\Gamma $. Then the product $D\otimes_{h} \Gamma$ is the digraph with vertex set $V(D)\times V$ and $((a,x),(b,y))\in E(D\otimes_h\Gamma)$ if and only if $(a,b)\in E(D)$ and $(x,y)\in E(h (a,b))$. In this paper, we introduce the undirected version of the $\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recover a generalization of the classical lexicographic product of graphs that was introduced by Sabidussi en 1961. We study connectivity properties and other invariants in terms of the factors. We also present a new intersection graph that emerges when we characterize the connectivity of $\otimes_h$-product of graphs.


Introduction
We begin by introducing those concepts of classical graph theory that will be necessary in this paper. First of all, we clarify that all the graphs considered in this paper are assumed to be finite and, if no otherwise specified, simple. Let G be a graph and let v ∈ V (G), we let the open neighborhood of v to be N G (v) = {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v is N G The degree of a vertex v, |N G (v)|, is denoted by d G (v) and the minimum degree among the vertices of G by δ(G). Let S be either a subset of V (G) or a subset of E(G), we denote by G[S] the subgraph of G induced by S. A set S ⊂ V (G) ∪ E(G) is a separating set if its deletion, which we denote by G − S, disconnects G. The minimum size of a separating set of vertices is called the connectivity of G, and is denoted by κ(G). The minimum size of a separating set of edges is called the edge-connectivity of G, and is denoted by λ(G). A separating set of vertices S is a κ-set if |S| = κ(G). Similarly, a λ-set is a separating set of edges of size λ(G). A set S ⊂ V (G) is a dominating set if each vertex in V (G) \ S is adjacent to at least one vertex of S. A dominating set S in which each vertex in S has a neighbor in S is called a total dominating set. The (total) domination number (γ t (G)) γ(G) of a graph G is the minimum cardinality of a (total) dominating set. The independence number of G, denoted by α(G) is the greatest r such that rK 1 , the complement of K r , is an induced subgraph of G. A maximal complete subgraph is a clique. The clique number ω(G) is the number of vertices of a maximum clique. An r-coloring of G is any function f : V (G) → {0, 1, 2, . . . , r − 1} such that if uv ∈ E(G) then f (u) = f (v). The chromatic number of G, χ(G), is the minimum r for which there is an r-coloring of G.
Let F = {S i : i ∈ I} be a family of sets. The intersection graph obtained from F is a graph that has a vertex v i for each i ∈ I, and for each pair i, j ∈ I, there is an edge v i v j if and only if S i ∩ S j = ∅.
Let G and H be two graphs. Two of the standard products of graphs are the direct and the lexicographic product. The direct product G ⊗ H (also denoted by G × H) is the graph with vertex set V (G) × V (H) and (a, x)(b, y) ∈ E(G ⊗ H) if and only if, ab ∈ E(G) and xy ∈ E(H). The direct product also appears in the literature as the cross product, the categorical product, the cardinal product, the tensor product, the relational product, the Kronecker product, the weak direct product and even the cartesian product. The lexicographic product G • H (also denoted by G [H]) is the graph with vertex set V (G) × V (H) and (a, x)(b, y) ∈ E(G • H) if and only if, either ab ∈ E(G) or a = b and xy ∈ E(H).
The following theorem is due to Weichsel.
Theorem 1.1. [15] Let G and H be graphs with at least one edge. Then G ⊗ H is connected if and only if both G and H are connected and at least one of them is nonbipartite. Furthermore, if both are connected and bipartite, then G ⊗ H has exactly two connected components. The lexicographic product of two graphs G and H, with G nontrivial, is connected if and only if G is connected.
Figueroa-Centeno et al. introduced the following product of digraphs in [3]: let D be a digraph and let Γ be a family of digraphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(D) −→ Γ. Then the product D⊗ h Γ is the digraph with vertex set V (D)×V and ((a, x), (b, y)) ∈ E(D⊗ h Γ) if and only if (a, b) ∈ E(D) and (x, y) ∈ E(h(a, b)). Notice that, when h is constant, the adjacency matrix of D ⊗ h Γ, A(D ⊗ h Γ), coincides with the classical Kronecker product of matrices, A(D) ⊗ A(h(e)), where e ∈ E(D). When |Γ| = 1, we refer to this product as the direct product of two digraphs and we just write D ⊗ Γ [16]. The ⊗ h -product of digraphs has been used to establish strong relations among different labelings and specially to produce (super) edge-magic labelings for some families of graphs [6,9,10]. Some structural results can be found in [1,9,10].
An undirected version of the ⊗ h -product can be provided as follows: let G be a graph and let Γ be a family of graphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(G) −→ Γ. Then the product G ⊗ h Γ is the graph with vertex set V (G) × V and (a, x)(b, y) ∈ E(G ⊗ h Γ) if and only if ab ∈ E(G) and xy ∈ E(h(ab)).
Motivated by the ⊗ h -product, we also recover a generalization of the lexicographic product that was introduced by Sabidussi in [17]. Let G be a graph and let Γ be a family of graphs. Consider any function h : V (G) −→ Γ. Then the product G • h Γ is the graph with vertex set ∪ a∈V (G) {(a, x) : x ∈ V (h(a))} and (a, x)(b, y) ∈ E(G • h Γ) if and only if either ab ∈ E(G) or a = b and xy ∈ E(h(a)).
The organization of the paper is the following one. Section 2 is dedicated to connectivity of both, the ⊗ h -product and the generalized lexicographic product. Section 3 is focused in the study of other invariants of the generalized products, in terms of the factors. We study the independence number, the domination number, the chromatic number and the clique number. We end up this paper by presenting some structural properties in Section 4.

Connectivity
Let G be a graph and let Γ be a family of graphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(G) −→ Γ. We denote by h(G) the graph with vertex set V and edge set The next result is, in some sense, a generalization of Theorem 1.1.
Theorem 2.1. Let G be a nontrivial connected graph and let Γ be a family of nontrivial connected graphs such that V (F ) = V for every F ∈ Γ. Consider any function h : Proof. First assume that G and h(G) are bipartite graphs with stable sets V (G) = A ∪ B and V = C ∪ D. Then, there are no edges between the sets of vertices Assume now that h(G) is nonbipartite. Since G is connected and δ(F ) ≥ 1, for every F ∈ Γ, in order to prove that G ⊗ h Γ is connected, we only have to prove that there exists a ∈ V (G) such that for each pair of vertices x, y ∈ V there is a path in G ⊗ h Γ connecting (a, x) and (a, y). If there exists e ∈ E(G) such that h(e) is nonbipartite, then by Theorem 1.1, the graph G[e] ⊗ h(e) is connected. Thus, we obtain that G ⊗ h Γ is connected. Suppose now that h(e) is bipartite, for each e ∈ E(G). By Theorem 1.1, the graph G[e] ⊗ h(e) has exactly two components.
Since all elements in Γ are connected and h(G) is nonbipartite, there exist a, b, c ∈ V (G), two elements of Γ, namely F 1 , F 2 , such that h(ab) = F 1 , h(bc) = F 2 , and the graph (V, E(F 1 )∪E(F 2 )) is nonbipartite. We denote by V i 1 , V i 2 the stable sets of F i , i = 1, 2. Claim. For each pair of vertices x, y ∈ V , there is a path connecting (a, x) and (a, y) in the subgraph Suppose that (a, x) and (a, y) are in different components of G[ab] ⊗ h(ab). Otherwise, the claim is trivial. Without loss of generality suppose that x ∈ V 1 1 and y ∈ V 1 2 . Assume first that V 1 k ∩ V 2 l = ∅, for each pair l, k ∈ {1, 2}, and let and (c, z), and also a path connecting (b, x ′ ) and (c, z). Therefore, there is a path connecting (a, x) and (a, y) in the subgraph of G ⊗ h Γ induced by {a, b, c} × V . Assume now that V 1 k ∩ V 2 l = ∅, for some pair l, k ∈ {1, 2}. Without loss of generality assume that V 1 1 ∩ V 2 1 = ∅. Then, we have V 1 2 ∩ V 2 2 = ∅. Otherwise, the graph (V, E(F 1 ) ∪ E(F 2 )) is bipartite, a contradiction. We then proceed as in the above case.
Assume now that h(G) is bipartite and that G is nonbipartite. Let C = a 0 a 1 . . . a 2k a 0 be an odd cycle in G. Since h(G) is bipartite, it follows that there exists a partition V = V 1 ∪ V 2 , such that V 1 and V 2 are the stable sets of h(a i a i+1 ) and h(a 2k a 0 ), for each i = 0, 1, . . . , 2k − 1. Which implies, since the cycle is odd, that we can connect every vertex in {a 0 } × V 1 to every vertex in {a 1 } × V 2 , {a 2 } × V 1 , and so on, until, {a 2k } × V 1 and finally, {a 0 } × V 2 . Therefore and by Remark 1.2, the graph G ⊗ h Γ is connected. ✷ When we do not assume that the elements of Γ are connected we can find examples of disconnected graphs that are of the form G ⊗ h Γ, with both G and h(G) nonbipartite and connected.
The next results give sufficient conditions to guarantee connectivity in G ⊗ h Γ when the family Γ contains disconnected graphs. The first result appears in the proof of Theorem 2.1. Notice that, if there exists ab ∈ V (G) such that h(ab) has an isolated vertex and either a or b is a vertex of degree 1 in G, then the graph G ⊗ h Γ is not connected. So, in what follows, we assume that all vertices of F have degree at least 1, for every F ∈ Γ. Recall that, since G is connected and δ(F ) ≥ 1, for every F ∈ Γ, in order to prove that G ⊗ h Γ is connected, we only have to prove that there exists a ∈ V (G) such that for every pair x, y ∈ V there is a path in G ⊗ h Γ connecting (a, x) and (a, y). This fact is guaranteed in the following two lemmas. Lemma 2.1. Let G be a nontrivial connected graph and let Γ be a family of graphs such that V (F ) = V and δ(F ) ≥ 1, for every F ∈ Γ. Consider any function h : E(G) −→ Γ. If there exists e ∈ E(G) such that h(e) is nonbipartite and connected. Then, the graph G ⊗ h Γ is connected.

Proof.
Let e ∈ E(G) such that h(e) is nonbipartite and connected. Then, by Theorem 1.1, the graph G[e] ⊗ h(e) is connected. ✷ Lemma 2.2. Let G be a nontrivial connected graph and let Γ be a family of graphs such that V (F ) = V and δ(F ) ≥ 1, for every F ∈ Γ. Consider any function h : E(G) −→ Γ. Let ab, bc ∈ E(G) such that h(ab) is bipartite and connected, with stable sets V 1 and V 2 and assume that at least one of the following holds: (i) One of the components of h(bc) is nonbipartite and contains vertices of V 1 and V 2 .
(ii) One of the components of h(bc) is bipartite, but one of the stable sets contains vertices of V 1 and V 2 .
Then, the graph G ⊗ h Γ is connected.
Proof. By Theorem 1.1, the subgraph G[ab] ⊗ h(ab) has two components, which are the subgraphs induced by the sets of vertices, is connected and contains vertices of the two components of G[ab] ⊗ h(ab). Therefore, all vertices of the form {a, b} × V are in the same component of G ⊗ h Γ and the result follows. (ii) Suppose now that C bc is a bipartite component of h(bc) which contains two vertices x, y ∈ V (h(bc)) in the same stable set, namely V 1 (C bc ), such that x ∈ V 1 and y ∈ V 2 . Thus, the subgraph of G ⊗ h Γ induced by Hence, the subgraph induced by {a, b} ⊗ V belongs to the same component of G ⊗ h Γ. Therefore, the result is proved. ✷ The next result is a technical theorem that presents an interesting relation between some properties of partitions and the connectivity of the intersection graph obtained from them.

Proof.
We denote each vertex of G with the name of the corresponding set of P 1 (A) ∪ P 2 (A) ∪ . . . ∪ P m (A). Let us see the sufficiency. Assume that, for each k are not in the same connected component of G.
Let us prove now the necessity. Suppose that G is disconnected, H is a connected component of G and let V (H) ∩ P i (A) = A i , for i = 1, 2, . . . , m. Clearly, we have that A i = P i (A), otherwise G is connected. We will prove that ∪ Ai∈Ai A i = ∪ Aj ∈Aj A j . We proceed by contradiction. Assume to the contrary that ∪ Ai∈Ai A i = ∪ Aj ∈Aj A j , for some pair i, j with 1 ≤ i ≤ j ≤ m. Without loss of generality assume that there exists a ∈ A i such that a / ∈ ∪ Aj ∈Aj A j . Thus, since we are dealing with For every graph F , we can associate a partition of V (F ), namely P F (V (F )) as follows. If H is a bipartite component of F , then each stable set of V (H) is an element of P F (V (F )). Otherwise, the set V (H) itself is an element of P F (V (F )).
Next, we are ready to state the following results. The proofs are a direct consequence of Theorem 2.3. The first result gives a characterization for the connectivity of G ⊗ h Γ when we relax the condition on connectedness over all elements of the family Γ, under the assumption that all connected components are nonbipartite.
Theorem 2.4. Let G be a nontrivial connected graph and let Γ be a family of graphs such that If we concentrate on the star then we can obtain a complete characterization, which does not depend on the bipartitness of the elements of Γ.
The above results give sufficient conditions that guarantee the connectivity of G⊗ h Γ, when Γ contains disconnected graphs. In order to provide a characterization, we introduce an intersection graph that we obtain from G, Γ and the function h : E(G) → Γ. Let G be a nontrivial connected graph and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h : Thus, by definition, we obtain the next characterization.
Theorem 2.6. Let G be a nontrivial connected graph and let Γ be a family of graphs such that Remark 2.7. It is worthly to mention that, as it happens with the direct product, the most natural setting for the ⊗ h -product is the class of graphs with loops. If we accept to apply our definition to graphs with loops, a rewiew of the proofs reveals that all results presented in this section remain valid in the class of graphs with loops.
2.1. Connectivity in G • h Γ. Let G be a graph and let Γ be a family of graphs. Consider any The next result is trivial. Lemma 2.3. Let G be a nontrivial graph and let Γ be a family of graphs. Consider any function When V (F ) = V for every F ∈ Γ, we can obtain exact formulas for the connectivity and the edgeconnectivity of G • h Γ, which generalize the ones corresponding to G • H. The proofs are similar to the case G • H in [14]. For each a ∈ V (G), the V -fiber of G • h Γ with respect to a, refers to Theorem 2.8. Let G be a connected graph of order n and let Γ be a family of graphs such that Proof.
) and the claim follows.
Suppose now that G is a connected graph of order n not isomorphic to K n , for n ≥ 1. Let S be a κ-set of G. Clearly, S × V is a separating set of G • h Γ and thus, κ(G • h Γ) ≤ κ(G)|V |. Consider now a separating set of vertices S of G • h Γ, we will show that |S| ≥ κ(G)|V |. We claim that there exist two vertices (a, x) and (b, y) that are in different connected components of G • h Γ − S, with a = b. Suppose to the contrary that G \ S ⊂ a V , for some a ∈ V (G). We obtain that |S| ≥ (n − 1)|V |, a contradiction with the fact that κ(G • h Γ) ≤ κ(G)|V | and G = K n . Since a = b there exist κ(G) disjoint paths in G, namely, P 1 , P 2 , . . . , P κ(G) connecting a and b. Let P = aa 1 a 2 . . . a r b be one of these paths. If for all a i there exists x i ∈ V such that (a i , x i ) / ∈ S, then (a, x) and (b, y) are connected through (a i , x i ), for i = 1, 2, . . . , r. Thus, for each path P j , there exists a i , such that {a i } × V ⊂ S. Hence, we have that |S| ≥ κ(G)|V | and the equality κ(G • h Γ) = |κ(G)||V | is proved when G = K n .✷ Notice that, the previous proof also shows that, when G = K n , a κ-set of G • h Γ is of the form ∪ a∈S a V , where S is κ-set of G. If we remove the hypothesis V (F ) = V , for every F ∈ Γ then, we cannot obtain this conclusion. However, the following inequality still holds: where the minimum is taken over all separating sets of vertices of G.
Theorem 2.9. Let G be a connected graph of order n ≥ 2 and let Γ be a family of nontrivial graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h :

Proof.
Clearly, all edges incident to a vertex (of minimum degree) form a separating set of edges. Similarly, from a λ-set S of G, we obtain a separating set {(a, x)(b, y) : ab ∈ S and x, y Let S be a λ-set of G • h Γ. Then, G • h Γ − S has exactly two connected components, namely C 1 and We can assume that A ∩ B = ∅, otherwise, A ∪ B is a partition of V (G) and, thus, the cardinality of the set of edges joining vertices of A with vertices of B is at least, λ(G). Hence, we obtain that |S| ≥ λ(G)|V | 2 , and the result follows.
] is a bipartite complete graph, and thus, with edge connectivity |V |.

Other invariants of generalized products
In this section we study some invariants related to the generalized products ⊗ h and • h . We start with the independence number. Based on Proposition 8.10 in [7], which is related to the independence number of the direct product, we have a clear lower bound for the independence number of G ⊗ h Γ. For each a ∈ V (G), the V -fiber of G ⊗ h Γ with respect to a, refers to a V = {(a, x) : x ∈ V } and the Proposition 3.1. Let G be a graph and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h : . Suppose now that I is an independent set of G, then ∪ a∈I a V is an independent set of G ⊗ h(G). Similarly, if J is an independent set of h(G) then ∪ x∈J G x is an independent set of G ⊗ h(G). Therefore, we get the result. ✷ With respect to the independence number in G • h Γ we can obtain an exact formula in terms of the independent sets of G. where the maximum is taken over all independent sets of vertices S of G.

Proof.
Let S be an independent set of vertices of G and let I a be a set of independent vertices of h(a). Then, the disjoint union ∪ a∈S {(a, x) : x ∈ I a } is an independent set of G • h Γ. Thus, we have that α(G • h Γ) ≥ max S a∈S α(h(a)). Suppose now that S • is a maximal independent set of vertices of G • h Γ and let S a = {x ∈ V (h(a)) : (a, x) ∈ S • }. For any pair a, b of vertices of G such that S a and S b are nonempty, we have that a and b are independent vertices. Moreover, the maximality of S • implies that if |S a | ≥ 1 then |S a | = |α(h(a))|. Hence, we obtain that |S • | = a∈S α(h(a)), where S is some independent set of vertices of V (G). Therefore, we have that α(G • h Γ) ≤ max S a∈S α(h(a)) and the result is proved.
, we obtain the next easy corollary. Recall that h(G) is the graph with vertex set V and edge set ∪ e∈E(G) E(h(e)).
Corollary 3.1. Let G be a graph and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h : Inspirated by Mekiš' lower bound proof, we improve the above lower bound for the domination number of the ⊗ h -product. We let h(G a ) = (V, ∪ b∈NG(a) E(h(ab))), for each a ∈ V (G).

Proof.
Notice that, if D ⊂ V (G) × V is a dominating set of G ⊗ h Γ, then π 1 (D) is a dominating set of G, where π 1 : V (G) × V → V (G) defined by π 1 (a, x) = a. In particular, γ(G) ≤ |π 1 (D)|. Similarly, for each a ∈ V (G), we can check that the set π 2 (∪ b∈NG[a] D ∩ b V ) is a dominating set of h(G a ), where π 2 is as defined in the proof of Theorem 2.8. Indeed, for every x ∈ V \ π 2 (∪ b∈NG[a] D ∩ b V ), the vertex (a, x) is adjacent to some (b, y) ∈ D. Thus, ab ∈ E(G) and xy ∈ E(h(ab)) and we obtain that Assume to the contrary that there exists a dominating set Similarly, if min a∈V (G) γ(h(G a )) = 1, the set π 1 (D) gives a dominating set of G of size at most γ(G) − 1, also a contradiction. Thus, we may assume that γ(G), γ(h(G a )) ≥ 2, for each a ∈ V (G). Let D 0 = {(a 1 , x 1 ), (a 2 , x 2 ), . . . , (a γ(G)−1 , x γ(G)−1 )} be a proper subset of D, with a i = a j , for each pair i, j, with i = j. Since |π 1 (D 0 )| = γ(G) − 1, there exists c ∈ V (G) \ π 1 (D 0 ) which is not adjacent to any of the vertices of π 1 (D 0 ). Consider now the set D \ D 0 . Since |D \ Notice that the vertex (c, y) is not adjacent to any vertex of D, which implies, since D is a dominating set that (c, y) ∈ D. Moreover, the condition c / ∈ π 1 (D 0 ) implies that, c ∈ D \ D 0 and that y ∈ π 2 (∪ b∈NG[c] (D \ D 0 ) ∩ b V ), a contradiction. ✷ In particular, since for each graph G, we have γ t (G) ≥ γ(G), we also obtain the following result.
Corollary 3.2. Let G be a graph and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h : The next generalization can be trivially obtained from Theorem 3.2.
Corollary 3.3. Let G and F be graphs and let Γ be a family of graphs such that V (F ′ ) = V (F ) = V and F is a subgraph of F ′ , for every F ′ ∈ Γ. Consider any function h : Proof.

Proof.
We let D ′ = ∪ e∈E(G) D e and B = ∪ e∈E(G) B e . We claim that X = (A × D ′ ) ∪ (D × B) is a dominating set of G ⊗ h Γ. We consider three cases. First, assume that a ∈ D \ A and x ∈ D ′ \ B. Since A is a dominating set of G there exists b ∈ A such that ab ∈ E(G). In particular, we have that x / ∈ B ab . Thus, there is y ∈ B ab such that xy ∈ E(h(ab)). Hence, we obtain that (b, y) ∈ X and (a, x)(b, y) ∈ E(G ⊗ h Γ). Assume now that a ∈ V (G) \ D and x ∈ V . Since A dominates G there is b ∈ A such that ab ∈ E(G). Now, the existence of y ∈ D ′ such that xy ∈ E(h(ab)) is guaranteed by considering the total dominating set D ab of h(ab). Finally, assume that (a, x) ∈ V (G) × (V \ D ′ ). Since D is a total dominating set there exists b ∈ D such that ab ∈ E(G). In particular, we have that x / ∈ B ab . Thus, there is y ∈ B ab such that xy ∈ E(h(ab)). Hence, we obtain that (b, y) ∈ X and (a, x)(b, y) ∈ E(G ⊗ h Γ).  The above upper bound is attained as shown the following example.
be any function that assigns F 1 to all of its edges except to one that receives F 2 . Then, since the graph However, it is not difficult to find examples in which the above upper bound it is not attained.
We have that h(G) ∼ = K 4 and since the graph contains an odd cycle (for instance, the subgraph generated by {(a, z), (b, y), (c, x)}, we obtain that 3 = χ( Related to the clique number, we have the following results. Lemma 3.4. Let G be a graph and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function h :

Proof.
Let {(a i , x i ) : i = 1, 2, . . . , k} be a maximal clique of G⊗ h Γ. By definition, we have that, a i a j ∈ E(G) and x i x j ∈ E(h(a i a j )). Thus, the sets {a i : i = 1, 2, . . . , k} and {x i : i = 1, 2, . . . , k} are complete subgraphs in G and h(G), respectively. ✷ Let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ, we denote by ΣΓ the graph with vertex set V and edge-set ∪ F ∈Γ E(F ).

Proof.
Let χ(G) = r and χ(h(a)) = s a , a ∈ V (G). Let g be an r-coloring of G and h a be an s a -coloring of h(a). We claim that f (a, x) = (g(a), h a (x)) defines a coloring of G • h Γ with at most r max v∈V (G) s v colors. Indeed, suppose that (a, x)(b, y) ∈ E(G • h Γ), then, either a = b and xy ∈ E(h(a)), which implies, since h a is an s a -coloring, that h a (x) = h a (y); or, ab ∈ E(G), but then, since g is an r-coloring of G, we have g(a) = g(b) and the result follows. ✷ The next examples show that the above upper bound is sharp and also, that there exist families of graphs for which the difference between the exact value and the upper bound can be arbitrarily large.
One of the main results found in the study of the chromatic number of the lexicographic product of graphs is the following one due to Geller and Sahl [4].
In the next lines we generalize Theorem 3.6 to the • h -product of graphs using similar ideas as the ones found in [4]. We first recall Proposition 1.20 in [7].

Proof.
Let v ∈ V (G) with χ(h(v)) = n. By Proposition 3.4 there exists a homomorphism f v : h(v) → K n . Thus, we can construct a homomorphism f from G • h Γ onto G • h ′ {K m } m∈N defined by f (a, x) = (a, f a (x)), for each (a, x) ∈ V (G • h Γ). Hence, again by Proposition 3.4, we have that Conversely, let f be an r-coloring of G • h Γ, with r = χ(G • h Γ). Let a ∈ V (G), the restriction to the set {a} × V (h(a)) contains at least n = χ(h(a)) colors. Choose n of them and a representative vertex in each color class. By connecting each pair of chosen vertices (in case they are not connected), eliminating the extra vertices and repeating the same process for every a ∈ V (G), we obtain an rcoloring of a graph which is isomorphic to By definition of the chromatic number, we obtain The reformulations that have been studied with respect to the chromatic number of the lexicographic product (see [7]) suggest new lines for future research. Suppose that we have a graph G, a function h : V (G) → Z + , and we assigne h(a) different colors from the set {0, 1, 2, . . . , s − 1} to each vertex a of G and adjacent vertices receive disjoint sets of colors. In that case, we say that the assignment is a h-tuple coloring. The h-chromatic number χ h (G) of G is the smallest s such that there is a h-tuple coloring with s colors. When h is constant and equal to n, then h-tuple coloring (and the h-chromatic number ) correspond to the n-tuple coloring (and the n th chromatic number) that was introduced by Stahl in [13].
Notice that, similar to what happens with the nth chromatic number, we have that And we also can establish a relation between χ(G • h ′ {K m } m∈N ) and Kneser graphs from a system of sets that we introduce in the following lines.
Let {r i } i∈I be a sequence of positive integers. Denote by K({r i } i∈I , s) the graph that has as vertex set the r i -subsets of a s-subset, for each i ∈ I, and two vertices are adjacent if and only if the subsets are disjoint. Clearly, each coloring a))}, where s is the number of colors used and h(a) = |V (h ′ (a))|. Moreover, for every homomorphism f : Thus, we get the next proposition.

Some structural properties
The next results can be though as some type of associative property for the two products, ⊗ h and • h .

4.1.
On the ⊗ h -decomposition for graphs. Notice that, each graph G admits a trivial decomposition in terms of the ⊗ h -product, namely G ∼ = L ⊗ G, where L denotes the graph with |V (L)| = |E(L)| = 1. We say that G has a nontrivial decomposition with respect the ⊗ h -product if there exist a graph H or order at least 2 (maybe with loops), a family of graphs Γ (maybe with loops), with V (F ) = V for every F ∈ Γ and a function h : E(H) → Γ such that G ∼ = H ⊗ h Γ. The next result gives necessary and sufficient conditions for the existence of nontrivial ⊗ h -decomposition for graphs.
Theorem 4.1. Let G be a graph. Then, G has a nontrivial decomposition with respect the ⊗ h -product if and only if there exists a partition V (G) = V 1 ∪ V 2 . . . ∪ V k , k ≥ 2, such that, for each i, j with 1 ≤ i ≤ j ≤ k, |V i | = |V j | and, there exist bijective functions ϕ i : V 1 → V i , such that, for each pair u, v ∈ V 1 , we have that Let us see the sufficiency. Assume that there exists a partition V (G) = V 1 ∪ V 2 . . . ∪ V k , k ≥ 2, such that, for each i, j with 1 ≤ i ≤ j ≤ k, |V i | = |V j | and, there exist bijective functions ϕ i : V 1 → V i , such that, for each pair u, v ∈ V 1 , we have that ϕ i (u)ϕ j (v) ∈ E(G) if and only if ϕ i (v)ϕ j (u) ∈ E(G). Let V 1 = {x s } l s=1 and let H be the graph with vertex set V (H) = {a 1 , a 2 , . . . , a k } and a i a j ∈ E(H) if and only if N G (V i ) ∩ V j = ∅, where N G (V i ) = ∪ v∈Vi N G (v). For every a i a j ∈ E(H), we consider the graph F ij with vertex set V 1 and edge set defined by x s x t ∈ E(F ij ) if and only if ϕ i (x s )ϕ j (x t ) ∈ E(G). Condition (4) guarantees that the graph F ij is well defined. Then, the bijective function f : V (G) → V (H)×V 1 defined by f (v) = (a i , ϕ −1 i (v)) if v ∈ V i , establishes an isomorphism between G and H ⊗ h Γ, where Γ = {F ij } aiaj ∈E(H) and h : E(H) → Γ is the function defined by h(a i a j ) = F ij . ✷ Notice that, if we require H to be a loopless graph then we can obtain a similar characterization only by adding the restriction on V i that says that V i is formed by independent vertices, for every i ∈ {1, 2, . . . , k}. Moreover, if we also require that the family Γ does not contain graphs with loops, then we should add the restriction ϕ i (u)ϕ j (u) / ∈ E(G), for each u ∈ V 1 and for each i, j with 1 ≤ i ≤ j ≤ k.
4.1.1. Non uniqueness. The next example shows, as it happens with the direct and the lexicographic products, that we do not have a unique decomposition in terms of the ⊗ h -product.
be a function in which F 1 is assigned to two edges and F 2 to the other edge. Then,