Functional Product of Graphs: Properties and applications

This paper presents a generalization of the cartesian product of graphs, which we call the functional product of graphs. We prove some properties of this new product, and we show that it is commutative, associative under certain conditions, and it has a neutral element, which consists of a single vertex without edges (the trivial graph). We present a characterization of the graphs, which can be obtained from functional product of other graphs. We prove that the maximum degree of the product graph is the sum of the maximum degrees of the factor graphs, and we present conditions that ensure the connectedness of the product graph. Finally, we present an application of the functional product of graphs, in which we prove some results that allow to generate graphs that admit an equitable total coloring, with at most ∆ + 2 colors.


Introduction
The cartesian product of graphs was first defined by Sabidussi [14] and Vizing [18] in the 1960's. Since then, a lot of work has been done on various topics related to the product graph. The product graphs have numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology [6]. Furthermore, the cartesian product presents some important algebraic properties. These properties were investigated, independently, by Sabidussi [14] and Vizing [18]. They showed that if we identify isomorphic graphs, then the cartesian product is commutative, associative, and it has a neutral element, which consists of a single vertex without edges (trivial graph).They also demonstrated that each connected finite graph has a decomposition into prime factors that is unique except for isomorphisms. Later, several works were done studying the behavior of the cartesian product with respect to the invariants of graphs. [6,1,8,13,17].
The total coloring was introduced independently by Behzad [2] and Vizing [19], and both conjectured that every graph admits a total coloring with at most ∆ + 2 colors. The total coloring of cartesian product of graphs has been investigated by different authors [6,12,15,16,22,23]. In [6], Kemnitz and Marangio investigated the total chromatic number of cartesian product of complete graphs, cycles, complete graphs and bipartite graphs, and cycles and bipartite graphs. In [15,16], the total chromatic number of the cartesian product of two paths, a path and a cycle, a path and a star, a cycle and a star, and two cycles are determined. Some partial results on the total coloring of cartesian products of several paths and several cycles are contained in [22]. In [23], Zmazek and Zerovnik generalized the result on [12], determining an upper bound for the total chromatic number of a graph.
Recently, Lozano et al. [9] have studied some relationships between equitable total coloring and range vertex coloring in some regular graphs. They proved that if a regular graph admits a 2-distant coloring with ∆ + 1 colors, then the coloring of the vertices can be completed to an equitable total coloring with at most ∆ + 2 colors. In [7] Lozano et al. showed the equivalence of a range coloring of order ∆ and the twodistant coloring [3]. These results motivated us to study the possibility of constructing families of regular graphs that admit a 2-distant coloring with ∆ + 1 colors.
In the section 3 of this paper, we introduce the concept of the functional product of graphs, and we show that it is a generalization of the cartesian product of graphs, and we prove some properties. In Section 4, we present an application of the functional product of graphs, and we prove some results that describe a method for obtaining harmonic graphs. We are going to show that all harmonic graphs admit an equitable total coloring with at most ∆ + 2 colors (i.e. it satisfies Wang's conjecture [20]). In this text, the graphs are simple, not oriented and without loops.

Basic Definitions and Notations
Below, we list the notations to be used throughout this paper: • {u, v} or uv denotes an edge of the graph G, in which u and v are adjacent; , if there is no ambiguity, denotes the degree of a vertex v in the graph G; • ∆(G) or ∆, if there is no ambiguity, denotes the maximum degree of the graph G; , if there is no ambiguity, denotes the set of all adjacent vertices to a vertex v in the graph G; • F (X) denotes the set of all bijections of X in X; • D(G) denotes the digraph obtained by replacing each edge uv of the graph G by arcs (u, v) and (v, u) while maintaining the same set of vertices; • D denotes the set of digraphs that satisfy the following conditions: is also an arc of this digraph.
2. No two arcs are alike.
denotes the graph obtained by replacing each pair of arcs (u, v) and (v, u) of − → G for the edge uv while maintaining the same set of vertices; • E(X) or E, if there is no ambiguity, denotes the set of edges (arcs) of the graph (digraph) X; • V (X) or V , if there is no ambiguity, denotes the set of vertices of the graph (digraph) X; be a set, k be a natural number, and C = {c 1 , c 2 , ..., c k } be an arbitrary set whose elements are called colors. A coloring of the graph G with the colors of C is an application f : S → C.
In the above definition, if S = V then f is a vertex coloring.
In the case that S = E, this is called an edge coloring. Fi-

Functional Product of Graphs
The main objective of this section is to present the definition of the functional product of graphs and to prove some properties of this new product. For this purpose, it is necessary to define applications, called linking applications, that associate each edge of a factor graph with a bijection defined on the set of vertices of another. This bijection indicates the manner in which the connection of the vertices of the product graph will be performed. We are also going to show also that the cartesian product of graphs can be viewed as a particular case of the functional product, in which all edges are associated to the identity application.
The applications f 1 and f 2 are called linking applications.
If D(G 1 ) and D(G 2 ) are functionally linked by applications , then the graphs G 1 (V 1 , E 1 ) and G 2 (V 2 , E 2 ) are said to be functionally linked through the same applications.
, is the digraph G * (V * , E * ) defined as follows: • (u, x), (v, y) ∈ E * if and only if one of the following conditions is true: Note that the cartesian product of graphs is a particular case of the functional product of graphs defined above, in which f 1 and f 2 assign the identity to all arcs of the corresponding digraphs. Figures 4 and 5 exemplify this relation.

Properties
It is immediate from definition 3.3 that if identify isomorphic graphs, then the functional product has neutral element, which consists of a single vertex without edges (the trivial graph). The following theorem shows that the functional product is commutative.
be graphs that are functionally linked by applications f 1 : In this sense, the functional product is commutative.
, we are going to prove that given two vertices (u, Applying the definition of functional product, we have that {(u, x), (v, y)} ∈ E * if and only if: Because 1 is equivalent to 4 and 2 is equivalent to 3, the theorem is proven.
The following result presents a characterization of the graphs that can be obtained from functional product of other graphs.

For all
Note that every graph has an orientation just replace each edge uv by exactly one and only one of the arcs (u, v) or (v, u).
The sets X 1 and X 2 are called matched if |X 1 | = |X 2 |, and there is a matching P ⊂ E, such that every edge of P has an end in X 1 and another in X 2 , and P saturates both X 1 and X 2 .
be simple graphs. There are linking applications f 1 : 2. For all edge uv ∈ E 1 , the sets {a((u, x)); x ∈ V 2 } and {a((v, y)); y ∈ V 2 } are matched. For each edge e ∈ E 1 , we denote by ε e the corresponding matching.
We defined: and only if one of the following conditions is satisfied: Initially, we are going to prove that G * is isomorphic to G.
On the other hand, because of 3, if {b((u, x)), b((v, y))} ∈ E, we have: So, G * is isomorphic to G.
It remains to prove that the applications f 1 and f 2 are linking applications. In fact, f 1 and f 2 satisfy conditions 1 and 2 of the linking application definition because of the way that they were defined. Now, if uv ∈ E 1 and xy ∈ E 2 are such that f 1 ((u, v))(x) = y and f 2 ((x, y))(u) = v, then the edge {(u, x), (v, y)} ∈ E * would be a double edge. It implies that G * (and therefore G) is not simple and this fact contradicts the hypotheses of the theorem. So, the applications f 1 and f 2 are linking applications.
Let suppose now that there are linking applications f 1 : We take the application a : } is a matching between the sets {a((u, x)); x ∈ V 2 } and {a((v, y)); y ∈ V 2 }. In a similar way, we have the matching ε xy for each xy ∈ E 2 .
To prove that E = ( , which is enough to prove the theorem.
See that the associativity of the cartesian product of graphs [2,14] is a consequence of the theorem 3.3 because if the bijections associated by the linking applications are always the identity, they satisfy the conditions of the theorem.

Invariants
In this section, we prove that the maximum degree of the product graph is the sum of the maximum degrees of the factor graphs, and we present conditions that ensure the connectedness of the product graph.
Proof. For each (u, x) ∈ V * , we call E G * ((u, x)) the set of edges that are incident on that vertex in the graph G * . The application h i : N G1 (u) → E G * ((u, x)) is constructed as follows. Let h 1 (v) = (v, y)(u, x), in which y ∈ V 2 is such that f 1 ((u, v))(x) = y, with (u, v) ∈ E(D(G 1 )), in which y exists because f 1 ((u, v)) is bijective. On the other hand, h 1 is injective because if v 1 , v 2 ∈ N G1 (u) and v 1 = v 2 , then necessarily (v 1 , y 1 )(u, x) = (v 2 , y 2 )(u, x) for any values of y 1 and y 2 . Similarly, we construct h 2 : N G2 (x) → E G * (u, x). If an edge is incident in (u, x) in the graph G * , then it has the form (u, x)(v, y). Then, it exists (u, v) ∈ E(D(G 1 )), such that f 1 ((u, v))(x) = y or (x, y) ∈ E(D(G 2 )) such that f 2 ((x, y))(u) = v. Due to construction h 1 and h 2 , we have that v))(x) = y and f 2 ((x, y))(u) = v. This contradicts condition 3 of the definition of linking applications, so it holds that h 1 (N G1 (u)) ∩ h 2 (N G2 (v)) = ∅. Now, we can construct the bijection as follows: This proves the theorem.
From the previous theorem, we immediately obtain the following corollary.  (V 1 , E 1 ) and G 2 (V 2 , E 2 ) be graphs that are functionally linked by applications f 1 : In general, the functional product of connected graphs is not necessarily connected, as it is showing in the next proposition.
Let's prove that the edge {(i, j), (i ′ , j ′ )} ∈ E * if and only if (i + j) and (i ′ + j ′ ) have the same parity. By the definition of functional product, {(i, j), (i ′ , j ′ )} ∈ E * if and only if one of the following conditions is true: In case 1, we have: If i is even and j is even, then i ′ is odd and j ′ is odd. If i is even and j is odd, then i ′ is odd and j ′ is even. If i is odd and j is even, then i ′ is even and j ′ is odd. If i is odd and j is odd, then i ′ is even and j ′ is even. In all cases, the sum has the same parity. In case 2, it is sufficient to proceed in a similar way to achieve the desired result. So, is disconnected and G * has 2 connected components of the same cardinality.
The following theorem gives a condition that ensures the connectedness of a functional product graph if the factors are connected. We are going to need two new concepts, namely centered applications and centroids.
Definition 3.8. Let G(V, E) be a graph, W be an arbitrary finite set, and f : E(D(G)) → F (W ) be an application, it is said that f is centered if it exists x ∈ W , such that f (e)(x) = x for all e ∈ E. Then, x is called a centroid of f . Theorem 3.5. Given two graphs G 1 (V 1 , E 1 ) and G 2 (V 2 , E 2 ) that are connected and functionally linked by applications f 1 : E(D(G 1 )) → F (V 2 ) and f 2 : E(D(G 2 )) → F (V 1 ), if f 1 or f 2 is a centered application, then the functional product G 1 by G 2 , with respect to f 1 and f 2 , is connected.
Proof. Without loss of generality, suppose that f 2 is centered, and let y ∈ E 1 be the centroid of f 2 , G * (V * , E * ) = (G 1 , f 1 ) × (G 2 , f 2 ), and V 2 = {u 1 , u 2 , . . . , u n }. Because y is the centroid and G 2 is connected, all vertices (y, u i ) ∈ V * , such that i ∈ {1, . . . n} are in the same connected component of G * . Now, let (x, u i0 ) ∈ V * be arbitrary, because G 1 is connected, there is a path This proves that all of the vertices of G * are in the same connected component. Therefore, G * is connected.
The cartesian product of graphs is connected if and only if both factors are connected. For more details, one can refer to [2,14]. Note that this result is a consequence of the theorem 3.5 because, in the cartesian product of graphs, the linking applications of f 1 and f 2 assign the identity to all arcs of the corresponding digraphs, ie, both are centered applications.

Applications Functional Product of Graphs
In this section, we present some results that show how generate harmonic graphs from any regular graph. As consequence of theorem 4.2, the total coloring of those graphs is equitable and, in consequence, it satisfies the Wang's Conjecture. In order to better understand the following results, we first state two theorems, which appears in [21] and [11] respectively.  |V (G)| is even, ∆(K n ) is odd and ∆(K n ) = ∆(G)+∆(G ′ ), it follows that ∆(G ′ ) is even. Initially suppose that ∆(G ′ ) is even, by Theorem 4.1, there is a decomposition of G ′ into two-factors. For each two-factors F , replace each cycle by an oriented cycle and define the application a : V (F ) −→ V (F ), such that if (u, v) ∈ E(F ), then a(u) = v and, clearly a is a bijection. The application f 2 associates the bijection a to each arc of the cycle and it associates the inverse bijection to each reverse oriented cycle. In the graph G, the application f 1 associates the identity to all pairs of arcs associated to edges. Now, if V (G) = v 0 , v 1 , v 2 , ..., v p , then in each vertex of the form (x, v p ), we apply the color p. By construction, the resulting coloring of G * = (G 1 , f 1 ) × (G 2 , f 2 ) is a coloring with range ∆ and it has ∆ + 1 colors. If ∆(G ′ ) is odd, then ∆(G) is even and so one only needs to change the positions of G and G ′ , in the previous reasoning, to obtain the desired result. Therefore, G * = (G, f 1 ) × (G ′ , f 2 ) is a harmonic graph.
Proof. It is only necessary to note that both G ′ and H can be decomposed in the same number of two-factors and each two-factor of G ′ has an associated bijection of vertices of G. Let F 1 , F 2 , F 3 , ..., F t be the two-factors of the decomposition of G ′ , let r 1 , r 2 , ..., r t be the associated bijections, and let K 1 , K 2 , k 3 , ..., K t be the two-factors of the decomposition of H, which will be replaced by oriented cycles O 1 , O 2 , ..., O t , the application of f 2 associates the bijections r i to each arc O i , and r −1 i to the reverse oriented cycle for all i ∈ 1, 2, ..., t. The application of f 1 associates the identity to all edges of G. Now, if V (G) = v 1 , v 2 , ..., v p , then in each vertex of the form (x, v p ), we apply the color p. Then, by construction, the resulting coloring of G * = (G 1 , f 1 ) × (G 2 , f 2 ) is a coloring with range ∆ and it has ∆ + 1 colors. Therefore, the graph is harmonic.  Figure 14 shows the equitable total coloring of the harmonic graph, obtained as a consequence of theorem 4.2. Figures 10, 11, 12 and 13 illustrate the proof of Theorem 4.4 using two cycles, C 5 and C 3 . Figure 14 shows the equitable total coloring of the harmonic graph, obtained as a consequence of theorem 4.2.

Conclusions
This paper presented the functional product of graph, which is a generalization of the cartesian product of graphs. We show that the functional product is commutative, it has a neutral element, and associative under certain conditions. We prove a result that offers a characterization of the product graphs, ie. it shows how are graphs that can be obtained by the functional product.
We studied some invariants. Initially, we proved that the maximum degree of the product graph is the sum of the maximum degrees of the factor graphs. In relation to connectedness, we showed that the functional product of connected graphs is not necessarily connected. We proved a result that gives some conditions in which the functional product of connected graphs is disconnected. In addition, we presented a condition that ensures the connectedness of a functional product graph if the factors are connected.
On the other hand, the functional product has proved to be efficient at constructing graphs that "inherit" desirable properties from the factors as was shown in Section 4. As application of the functional product, we proved two theorems that ensure that harmonic graphs can be constructed using the functional product of graphs and any regular graphs as basis.
In future work, it will be studied the behavior of other invariants of graphs, for example chromatic number, connectivity, dominance, and diameter. In addition, it will be studied the possibility of recognizing families or subfamilies of graphs that can be obtained by the functional product. For example, the figures below 15 and 16 illustrate the Kneser graph KG 5,2 isomorphic to the Petersen Graph generated by the functional product of a P 2 and a C 5 .     Figure 11: Cycle obtained from the decomposition of two-factors from the graph G ′ with an arbitrary orientation.