Connectivity and other invariants of generalized products of graphs

Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma}$$\end{document} be a family of digraphs such that V (F) = V for every F∈Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F \in \Gamma}$$\end{document}. Consider any function h:E(D)→Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h : E(D) \rightarrow \Gamma}$$\end{document}. Then the product D⊗hΓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D \otimes_h \Gamma}$$\end{document} is the digraph with vertex set V(D)×V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V (D) \times V}$$\end{document} and ((a,x),(b,y))∈E(D⊗hΓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${((a, x), (b, y)) \in E (D \otimes_h \Gamma)}$$\end{document} if and only if (a,b)∈E(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(a, b) \in E (D)}$$\end{document} and (x,y)∈E(h(a,b))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(x, y) \in E (h (a, b))}$$\end{document}. In this paper, we deal with the undirected version of the ⊗h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\otimes_h}$$\end{document}-product, which is a generalization of the classical direct product of graphs and, motivated by the ⊗h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\otimes_h}$$\end{document}-product, we also recover a generalization of the classical lexicographic product of graphs, namely the ∘h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\circ_h}$$\end{document}-product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G⊗hΓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G \otimes_h \Gamma}$$\end{document} that generalize the existing one for the direct product. For G∘hΓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G \circ_h \Gamma}$$\end{document}, we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F∈Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F \in \Gamma}$$\end{document}. We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ⊗h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\otimes_h}$$\end{document}-product and the ∘h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\circ_h}$$\end{document}-product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ⊗h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\otimes_h}$$\end{document}-product.


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, , 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 [19]. 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 [8,10,11]. Some structural results can be found in [1,10,11].
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) . Using the ⊗ h -product of graphs, we can enlarge the set of applications to other types of labelings. For instance, we can obtain distance magic labelings [13,17] of regular graphs that come from this product. Due to the links existing with the world of graph labelings, we find interesting to study connectivity and other invariants of the graphs obtained by the ⊗ h -product.
Motivated by the ⊗ h -product, we also recover a generalization of the lexicographic product that was introduced by Sabidussi in [15], named by him Y -join. Let G be a graph and let Γ be a family of graphs. Consider any 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 E h(G) = ∪ e∈E(G) E h(e) . Clearly, if (a, x) ∈ V (G ⊗ h Γ) then 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 Since all elements in Γ are connected and h(G) is nonbipartite, there exist e, f ∈ E(G), two elements of Γ, namely F 1 , F 2 , such that h(e) = F 1 , h(f ) = F 2 , and the graph V, E(F 1 ) ∪ E(F 2 ) is nonbipartite. It is a fact that e and f can be chosen to be adjacent. Otherwise, since G is connected, there exists a path e = e 1 , e 2 , . . . , e r = f such that h(e 1 ) and h(e 2 ) have the same stable sets, since e 1 and e 2 are adjacent. Reasoning inductively in a similar way, we obtain that h(e) = h(e 1 ) and h(f ) = h(e r ) have the same stable sets, which contradicts that the graph V, Claim. For each pair of vertices x, y ∈ V , there is a path connecting (a, x) and (a, y)  x , y ∈ V 2 2 , that for each z ∈ V 2 1 there is a path in G[bc] ⊗ F 2 connecting (b, y ) 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 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. 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, (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 with 1 k m, there exists A k ⊂ P k (A), such that 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 partitions of A, there exists B j ∈ P j (A) \ A j such that a ∈ B j , and hence, For every graph F , we can associate a partition of V (F ), namely 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 the connectivity 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 bipartiteness 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 V (F ) = V , for every F ∈ Γ. Consider any func- Remark 2.7. It is worthy 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 (see [7] for an extensive discussion about the role of loops in direct products). If we accept to apply our definition to graphs with loops, a review of the proofs reveals that all results presented in this section remain valid in the class of graphs with loops.

Connectivity in G • h Γ. Let G be a graph and let Γ be a family of graphs. Consider any function
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 edge-connectivity of G • h Γ, which generalize the ones corresponding to G • H. The proofs for the connectivity and for the edgeconnectivity are similar to the case G • H in [7] (see Proposition 25.7 and Exercise 25.8) and [20], respectively. For each a ∈ V (G), the V -fiber of G • h Γ with respect to a, refers to a V = (a, x) : x ∈ V . Theorem 2.8. Let G be a connected graph of order n and let Γ be a family of graphs such that V (F ) = V , for every F ∈ Γ. Consider any function Proof. Suppose first that G = K n , for n 1. We claim that Consider 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 Notice that, the previous proof also shows that, when 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.
GENERALIZED PRODUCTS OF GRAPHS S. C. LÓPEZ and F. A. MUNTANER-BATLE 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 : V (G) → Γ. Then 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) Let S be a λ-set of G • h Γ. Then, G • h Γ − S has exactly two connected components, namely C 1 and C 2 . Consider the subsets A = a ∈ V (G) : 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. Let ] is a bipartite complete graph, and thus, with edge connec- has connectivity |V |, we have that (2) |S ab | |V |, for all b ∈ N G (a).

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 Propo-S. C. LÓPEZ and F. A. MUNTANER-BATLE sition 27.11 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 G-fiber of G ⊗ h Γ with respect to x ∈ V is G x = (a, x) : a ∈ V (G) .
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 : E(G) → Γ.
. 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.

Proposition 3.2. Let G be a graph of order n 2 and let Γ be a family of graphs. Consider any function
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 , 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.

Domination number.
One of the famous conjectures on graphs is Vizing's conjecture on the domination number of Cartesian products of graphs (see [2] for a recent survey on it). Although Gravier and Khellady [6] posed a kind of Vizing's conjecture for the direct product of graphs, namely γ(G ⊗ H) γ(G)γ(H), a year later Nowakowski and Rall [14] gave GENERALIZED PRODUCTS OF GRAPHS 12 S. C. LÓPEZ and F. A. MUNTANER-BATLE a counterexample. In fact, Klavžar and Zmazek [9] showed that the difference γ

(G)γ(H) − γ(G ⊗ H) can be arbitrarily large. Recently, Mekiš has shown in [12] that for arbitrary graphs G and H, we have γ(G ⊗ H) γ(G) + γ(H) − 1. Thus, since E(G ⊗ h Γ) ⊂ E G ⊗ h(G)
, 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 : Inspired 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).

Theorem 3.1. Let G be a graph and let Γ be a family of graphs such that
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).
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 S. C. LÓPEZ and F. A. MUNTANER-BATLE

13
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
With respect the upper bound for the domination number of direct product of graphs, Brešar et al. proved in [3] the next theorem.

Proof. Let us consider the inclusion E(G ⊗ F ) ⊂ E(G ⊗ h Γ). Thus, we have that γ(G ⊗ h Γ) γ(G ⊗ F ) and the result follows from Theorem 3.2.
If the existence of a spanning connected subgraph F for every graph F in Γ is not assumed in Γ, then, we cannot control the size of a dominating set of G ⊗ h Γ. However, a similar proof as the one of Theorem 3.2 in [3] allows us to obtain the next result. Proof. We let D = ∪ e∈E(G) D e and B = ∪ e∈E(G) B e . We claim that 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) . 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).  D a∈D γ h(a) , where the minimum is taken over all dominating sets D of G.

The chromatic number and the clique number.
Similarly to the independence number and based on the inequality χ(G ⊗ H) min χ(G), χ(H) (see for instance, [7]) we get the next trivial lemma.
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 Related to the clique number, we have the following results. 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 ).    r and χ h(a) = s a , a ∈ V (G). Let g be an rcoloring 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.   let {0, 1, . . . , n − 1} and {n, n + 1} be the colors assigned to (a, x) : x ∈ V (K n ) and (b, y) : y ∈ V (K 2 ) respectively. By assigning 1, n and n + 1 to {c} × V (2K 1 ), {d} × V (2K 1 ) and {e} × V (2K 1 ), respectively, we obtain a (n + 2)-coloring of G • h Γ.
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 Stahl [5].
In the next lines we generalize Theorem 3.6 to the • h -product of graphs using similar ideas as the ones found in [5]. We first recall Proposition 2.10 in [7].   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 r-coloring of a graph which is isomorphic 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 [16].
Notice that, similar to what happens with the n th 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,

Some structural properties
The next results can be thought as some type of associative property for the two products, ⊗ h and • h .  (α, a)(β, b) = h(ab). Then, we have that V G ⊗ (H ⊗ h Γ) = V (G ⊗ H) ⊗ h Γ and the identity function between the sets of vertices defines an isomorphism of graphs. Indeed, (α, a), x (β, b) αβ ∈ E(G) and ab ∈ E(H) (ii) For all h : V (G • H) → Γ there exists a family Γ and a function h :  h(α, a). Consider now, the function h : V (G) → Γ defined by, h (α) = H • hα Γ. Then, an easy check shows that V (G • H) • h Γ = V (G • h Γ ) and the identity function between the sets of vertices defines an isomorphism of graphs.

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 Proof. Assume that there exist a nontrivial graph H, a family of graphs Γ, with V (F ) = V for every F ∈ Γ and a function h : E(H) → Γ such that G ∼ = H ⊗ h Γ. Clearly the V -fibers of H ⊗ h Γ form a partition of V (G), namely ∪ a∈V (H)a V . Let a ∈ V (H). For any vertex b of H, consider the function ϕ b defined by ϕ b (a, x) = (b, x). Then, by definition of the ⊗ h -product, we have that (b, x)(c, y) ∈ E(H ⊗ h Γ) if and only if (b, y)(c, x) ∈ E(H ⊗ h Γ). Thus, condition (4) holds.
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 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 : 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,