(Di)graph decompositions and magic type labelings: a dual relation

A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence of $f$. An edge-magic labelling with the extra property that $f(V(G))= \{1,2,\ldots,|V(G)|\}$ is called super edge-magic. In this paper, we establish a relationship between the valences of (super) edge-magic labelings of certain types of bipartite graphs and the existence of a particular type of decompositions of such graphs.


Introduction
For the graph theory terminology and notation not defined in this paper we refer the reader to either one of the following sources [2,3,8,13,21]. However, in order to make this paper reasonably self-contained, we mention that by a (p, q)-graph we mean a graph of order p and size q. For integers m ≤ n, we use [m, n] to denote the set {m, m + 1, . . . , n}. In 1970, Kotzig and Rosa [12] introduced the concepts of edge-magic graphs and edge-magic labelings as follows: Let G be a (p, q)-graph. Then G is called edge-magic if there is a bijective function f : V (G) ∪ E(G) → [1, p + q] such that the sum f (x) + f (xy) + f (y) = k for any xy ∈ E(G). Such a function is called an edge-magic labeling of G and k is called the valence [12] or the magic sum [21] of the labeling f . We write val(f ) to denote the valence of f .
Motivated by the concept of edge-magic labelings, Enomoto et al. [4] introduced in 1998 the concepts of super edge-magic graphs and labelings as follows: Let f : V (G) ∪ E(G) → [1, p + q] be an edge-magic labeling of a (p, q)-graph G with the extra property that f (V (G)) = [1, p]. Then G is called super edge-magic and f is a super edge-magic labeling of G. We take this opportunity to mention that although the definitions of (super) edge-magic graphs and labelings were originally provided for simple graphs (that is to say, graphs with no loops nor multiple edges), in this paper, we understand these definitions for any graph. Therefore, unless otherwise specified, the graphs considered in this paper are not necessarily simple. In [5], Figueroa-Centeno et al. provided the following useful characterization of super edge-magic simple graphs, that works in exactly the same way for graphs in general. Lemma 1.1. [5] Let G be a (p, q)-graph. Then G is super edge-magic if and only if there is a bijective function g : V (G) −→ [1, p] such that the set S = {g(u) + g(v) : uv ∈ E(G)} is a set of q consecutive integers. In this case, g can be extended to a super edge-magic labeling f with valence p + q + min S.
Unless otherwise specified, whenever we refer to a function as a super edge-magic labeling we will assume that it is a function f as in Lemma 1.1. Before moving on, it is worthwhile mentioning that Acharya and Hegde had already defined in 1991 [1] the concept of strongly indexable graphs. This concept turns out to be equivalent to the concept of super edge-magic graphs. However in this paper we will use the names super edge-magic graphs and super edge-magic labelings. In [7] Figueroa et al., introduced the concept of super edge-magic digraph as follows: a digraph D = (V, E) is super edge-magic if its underlying graph is super edge-magic. In general, we say that a digraph D admits a labeling f if its underlying graph admits the labeling f . It was also in [7] that the following product was introduced: let D be a digraph and let Γ be a family of digraphs with the same set V of vertices. Assume that h : E(D) → Γ is any function that assigns elements of Γ to the arcs of D.
Many relations among labelings have been established using the ⊗ h -product and some particular families of graphs, namely S p and S k p (see for instance, [11,16,18,20]). The family S p contains all super edge-magic 1-regular labeled digraphs of order p where each vertex takes the name of the label that has been assigned to it. A super edgemagic digraph F is in S k p if |V (F )| = |E(F )| = p and the minimum sum of the labels of adjacent vertices is equal to k (see Lemma 1.1). Notice that, since each 1-regular digraph has minimum induced sum equal to (p + 3)/2, S p ⊂ S (p+3)/2 p . The following result was introduced in [18], generalizing a previous result found in [7] : Theorem 1.1. [18] Let D be a (super) edge-magic digraph and let h : E(D) → S k p be any function. Then und(D ⊗ h S k p ) is (super) edge-magic. Remark 1.1. The key point in the proof of Theorem 1.1 is to rename the vertices of D and each element of S k p after the labels of their corresponding (super) edge-magic labeling f and their super edge-magic labelings respectively. Then the labels of the product are defined as follows: (i) the vertex (a, i) ∈ V (D ⊗ h S k p ) receives the label: p(a − 1) + i and (ii) the arc ((a, i), (b, j)) ∈ E(D ⊗ h S k p ) receives the label: p(e − 1) + (k + p) − (i + j), where e is the label of (a, b) in D. Thus, for each arc ((a, i), (b, j)) ∈ E(D ⊗ h S k p ), coming from an arc e = (a, b) ∈ E(D) and an arc (i, j) ∈ E(h(a, b)), the sum of labels is constant and equal to p(a+b+e−3)+(k+p). That is, p(val(f )−3)+k+p. Thus, the next result is obtained.
All the results in the literature involving the ⊗ h -product had super edge-magic labeled digraphs in the second factor of the product. However, in [14] it was shown that other labeled (di)graphs can be used in order to enlarge the results obtained, showing that the ⊗ h -product is a very powerful tool. Next, we introduce the family T q σ of edge-magic labeled digraphs. An edge-magic labeled digraph F is in T q σ if V (F ) = V , |E(F )| = q and the magic sum of the edge-magic labeling is equal to σ.
we assign to the vertex the label: we assign to the arc the label: (p + q)(k + n − (i + j) − 1) + (σ − (a + b)). Notice that, since D ∈ S k n is labeled with a super edge-magic labeling with minimum sum of the adjacent vertices equal to k, the sum of the labels is constant and is equal to: In [17] López et al. introduced the following definitions. Let G = (V, E) be a (p, q)-graph. Then the set S G is defined as then the super edge-magic interval of G, denoted by I G , is defined to be the set I G = [⌈min S G ⌉, ⌊max S G ⌋] and the super edge-magic set of G, denoted by σ G , is the set formed by all integers k ∈ I G such that k is the valence of some super edge-magic labeling of G. A graph G is called perfect super edge-magic if σ G = I G . In order to conduct our study in this paper, the following lemma will be of great help.
The graph formed by a star K 1,n and a loop attached to its central vertex, denoted by K l 1,n , is perfect super edge-magic for all positive integers n. Furthermore, |I K l 1,n | = |σ K l 1,n | = n + 1. In [19] the same authors generalized the previous definitions to edge-magic graphs and labelings as follows: Let G = (V, E) be a (p, q)-graph, and denote by T G the set If ⌈min T G ⌉ ≤ ⌊max T G ⌋ then the magic interval of G, denoted by J G , is defined to be the set J G = [⌈min T G ⌉, ⌊max T G ⌋] and the magic set of G, denoted by τ G , is the set τ G = {n ∈ J G : n is the valence of some edge-magic labeling of G}. It is clear that In the next lemma, we provide a well known result that gives a lower bound and an upper bound for edge-magic valences. We add the proof as a matter of completeness. Recall that the complementary labeling of an edge-magic labeling f is the labeling Proof.
be an edge-magic labeling of G. The two lowest possible integers in [1, p + q − 1] that can be added to p + q are 1 and 2. Thus, val(f ) ≥ p+q +3. By using the complementary labeling, the maximum possible valence has the form 3 The study of the (super) edge-magic properties of the graph C m ⊙ K n as a particular subfamily of S k n has been of interest recently. See for instance [15,17,19]. Due to this, many things are known on the (super) edge-magic properties of the graphs C p k ⊗ K n [19] and C pq ⊗ K n [15], where p and q are coprime. However, many other things remain a mystery, and we believe that it is worth the while to work in this direction. In fact, a big hole in the literature, appears when considering graphs of the form C m ⊙ K n for m even. In this paper, we will devote Section 2 to this type of graphs. This study leads us to consider other classes of graphs and to study the relation existing between the valences of edge-magic and super edge-magic labelings and the well known problem of graph decompositions.
A decomposition of a simple graph G is a collection We want to bring this introduction to its end by saying that the interested reader can also find excellent sources of information about the topic of graph labeling in [2,8,10,13,21].

More valences
As we have already mentioned in the introduction, not too much is known about the valences of (super) edge-magic labelings for the graph C m ⊙ K n when m is even. In fact, as far as we know, the only papers that deal with (super) edge-magic labelings of C m ⊙ K n for m even are [6,15]. Hence almost all such results involve only odd cycles. Next, we study the edge-magic valences of C m ⊙ K n when m is even. Unless otherwise specified, − → G denotes any orientation of G. The next lemma is an generalization of Lemma 12 in [15].
Lemma 2.1. Let g be a (super) edge-magic labeling of a graph G, and let f r be the super edge-magic labeling of K l 1,n that assigns label r to the central vertex, 1 ≤ r ≤ n+1. Then, (i) the induced (super) edge-magic labeling g r of − → G ⊗ − → K l 1,n has valence (n+1)(val(g)− 2) + r + 1.
(ii) Let g ′ be a different (super) edge-magic labeling of G with val(g) < val(g ′ ), then val( g n+1 ) < val( g ′ 1 ), where g ′ r is the induced (super) edge-magic labeling of − → G ⊗ − → K l 1,n when K l 1,n is labeled with f r and G with g ′ .
By adding an extra condition on the smallest and the biggest valence, we can improve the lower bound given in the previous result.
Proof. Let G = C m 1 ⊕ C m 2 ⊕ · · · ⊕ C m k and let − → G = C + m 1 ⊕ C + m 2 ⊕ · · · ⊕ C + m k be an orientation of G in which each cycle is strongly oriented. Then n ). Note that since G is bipartite, all cycles should be of even length and by definition of ⊗-product, G ⊙ K n ∼ = und( − → G ⊗ − → K l 1,n ). Thus by Theorem 2.1, |τ G⊙Kn | ≥ (n + 1)|τ G | + 2.
. Moreover, since the condition val(β) − val(α) < (val(α) − (p + q + 2))n, is satisfied for n ≥ 2, by using Theorem 1.2, val( g r ) = 8(1 + r) + val(g) which gives, associated to a labeling g two new valences, namely val( g 1 ) and val( g 3 ) which gives in total 20 valences. The induced labelings and they are shown in Fig. 1, according to the notation introduced above (for clarity reasons, only the labels of the vertices are shown). Notice that, by using the missing labels, there is only one way to complete the edge-magic labelings obtained in Fig.  1. The minimum induced sum together with the maximum unused label provides the valence of the labeling.
It is well known that all cycles are edge-magic [9]. Thus, the following corollary follows: A similar argument to that of the first part in Theorem 2.1 can be used to prove the following theorem.

A relation between (super) edge-magic labelings and graph decompositions
Let G be a bipartite graph with stable sets X = {x i } s i=1 and Y = {y j } t j=1 . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then we denote by S 2 (G; H 1 , H 2 ) the graph with vertex and edge sets defined as follows: We are ready to state and prove the next theorem.
Theorem 3.1. Let G be a bipartite (super) edge-magic simple graph with stable sets X and Y . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then, the graph S 2 (G; H 1 , H 2 ) is (super) edge-magic.
Proof. Let f be a (super) edge-magic labeling of G, and assume that the edges of H 1 are directed from X to Y and the edges of Moreover, an easy check shows that the bijective function φ : . Therefore, the graph S 2 (G; H 1 , H 2 ) is (super) edge-magic.
Next, we show an example.   Kotzig and Rosa [12] proved that every complete bipartite graph is edge-magic. It is clear that Theorem 3.1 works very nicely when the graph G under consideration is a complete bipartite graph and many new edge-magic graphs can be obtained. Theorem 3.1 can be easily extended. Let us do so next.
Let G be a bipartite graph with stable sets X = {x i } s i=1 and Y = {y j } t j=1 . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then we define S 2n (G; H 1 , H 2 ) to be the graph with vertex and edge sets as follows: Let G be a bipartite simple graph with stable sets X and Y . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then, there exists an orientation of G and K l 1,n , namely − → G and − → K l 1,n respectively, such that S 2n (G; }. An easy check shows that the bijective function φ : We are ready to state and prove the next theorem. Theorem 3.2. Let G be a bipartite (super) edge-magic simple graph with stable sets X and Y . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then, the graph S 2n (G; H 1 , H 2 ) is (super) edge-magic.
Proof. Let f be a (super) edge-magic labeling of G, and assume that the edges of H 1 are directed from X to Y and the edges of H 2 are directed from Y to X in G, obtaining the digraph − → G . Let − → K l 1,n be the super edge-magic labeled digraph with V ( With the help of Lemma 1.3, we can generalize Theorem 3.2 very easily. We do it in the following two results. Theorem 3.3. Let G be a bipartite super edge-magic simple graph with stable sets X and Y . Assume that G admits a decomposition G ∼ = H 1 ⊕ H 2 . Then |σ S 2n (G;H 1 ,H 2 ) | ≥ (n + 1)|σ G |.
Proof. Let h be a super edge-magic labeling of G, and assume that the edges of H 1 are directed from X to Y and the edges of H 2 are directed from Y to X in G, obtaining the digraph − → G . Let f r be the super edge-magic labeling of − → K l 1,n that assigns the label r to the central vertex with val(f r ) = 2n + 3 + r, 1 ≤ r ≤ n + 1. Then by Lemma 3.1, S 2n (G; H 1 , H 2 ) ∼ = und( − → G ⊗ − → K l 1,n ) and by Theorem 3.2, it is super edge-magic. By Theorem 2.3, |σ S 2n (G;H 1 ,H 2 ) | ≥ (n + 1)|σ G |.
A similar argument to the one of Theorem 3.3, but now using Theorem 2.1, allows us to prove the following theorem.
Once again, we have the following two easy corollaries.  At this point, consider any graph G * whose vertex set admits a partition of the form V (G * ) = X ∪ Y ∪ n k=1 X k ∪ n k=1 Y k and that decomposes as a union of three bipartite graphs G * ∼ = G⊕H 1 ⊕H 2 , where G * [X ∪Y ] ∼ = G, G * [X ∪Y k ] ∼ = H 1 and G * [X k ∪Y ] ∼ = H 2 for all k ∈ [1, n]. By Theorem 3.1, we have the following remarks.
Remark 3.1. If G is a (super) edge-magic graph and G * is not, then H 1 and H 2 do not decompose G.
We will bring this section to its end, by mentioning that, although some labelings involving differences as for instance, graceful labelings and α-valuations have a strong relationship with graph decompositions, the results mentioned in this section are the only ones known relating the subject of decompositions with addition type labelings. This is why we consider these results interesting.

Conclusions
The goal of this paper is to show a new application of labeled super edge-magic (di)graphs to graph decompositions. The relation among labelings and decompositions of graphs is not new. In fact, one of the first motivations in order to study graph labelings was the relationship existing between graceful labelings of trees and decompositions of complete graphs into isomorphic trees. What we believe that it is new and surprising about the relation established in this paper is that, as far as we know, there are no relations between labelings involving sums and graph decompositions. In fact, we believe that this is the first relation found in this direction and we believe that to explore this relationship is a very interesting line for future research.