Perfect (super) edge-magic crowns

In this paper we continue the study of the valences for (super) edge-magic labelings of crowns $C_{m}\odot \overline K_{n}$ and we prove that the crowns are perfect (super) edge-magic when $m=pq$ where $p$ and $q$ are different odd primes. We also provide a lower bound for the number of different valences of $C_{m}\odot \overline K_{n}$, in terms of the prime factors of $m$.


Introduction
For the graph theory terminology and notation not defined in this paper we refer the reader to either one of the following sources [2,3,7,17]. 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. In 1970, Kotzig and Rosa [10] 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) → {i} p+q i=1 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 [10] or the magic sum [17] 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) → {i} p+q i=1 be an edge-magic labeling of a (p, q)-graph G with the extra property that f (v) = {i} p i=1 . Then G is called super edge-magic and f is a super edge-magic labeling of G. It is worthwhile mentioning that Acharya and Hegde had already defined in [1] the concept of strongly indexable graph that turns out to be equivalent to the concept of super edge-magic graph. We take this opportunity to mention that although the original 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) −→ {i} p i=1 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. Enomoto et al. [4] were the first ones to observe the following result for which we provide the proof as a matter of completeness.

Lemma 1.2. [4]
A cycle of order n is super edge-magic when n is odd.
We will refer to the labeling introduced in the proof of the previous lemma as the canonical labeling of the cycle. When we say that a digraph has a labeling we mean that its underlying graph has such labeling, see [6]. We denote the underlying graph of a digraph D by und(D). Remark 1.1. Let {f (C n ) + , f (C n ) − } be the strong orientations of the super edge-magic labeled cycle C n introduced in the proof of Lemma 1.2. Then 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 ⌋] ∩ Z 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 A famous conjecture of Godbold and Slater [8] states that, for n = 2t + 1 ≥ 7 and 5t + 4 ≤ j ≤ 7t + 5 and for n = 2t ≥ 4 and 5t + 2 ≤ j ≤ 7t + 1 there is an edge-magic labeling of C n , with valence k = j. That is, for odd n ≥ 7 and for even n ≥ 4 the cycle C n is perfect edge-magic.
Let G = (V, E) be a (p, q)-graph. Then the set S G is defined as 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 graph [12] if σ G = I G .
Lemma 1.3. 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.
Proof. By Lemma 1.1, it is a very easy observation that any bijection f : is a super edge-magic labeling of K l 1,n . Further more, the valence of any super edge-magic labeling of K l 1,n depends only on the label assigned to the central vertex of K l 1,n (that is, the vertex of K l 1,n with degree different from 1). If two labelings of K l 1,n assign consecutive labels to the central vertex of K l 1,n , then the resulting valences are also consecutive. Since there are exactly (n + 1) possible consecutive labels to assign to the central vertex, it follows that |I K l 1,n | = |σ K l 1,n | = n + 1.
The corona product of two graphs G and H is the graph G ⊙ H obtained by placing a copy of G and |V (G)| copies of H and then joining each vertex of G with all vertices in one copy of H in such a way that all vertices in the same copy of H are joined exactly to one vertex of G. Let K n be the complementary graph of the complete graph K n , n ∈ N. Theorem 1.1. [12,13] Let C m be a cycle of order m = p k , where p > 2 is a prime number. Then the graph G ∼ = C m ⊙ K n is perfect (super) edge-magic.
In this paper, we extend the result to m = pq, where p and q are different odd primes. The paper is organized as follows: in Section 2, we provide all the necessary results needed for this paper. In Section 3, we prove that each element in the family C m ⊙ K n where m = pq, with p and q being different odd primes is a perfect (super) edge-magic graph. In Section 4, we provide a lower bound for the number of valences of general crowns C m ⊙ K n .

The tools
Let f be an edge-magic labeling of a (p, q)-graph G. The complementary labeling of f , denoted by f , is the labeling defined by the rule: Notice that, if f is an edge-magic labeling of G, we have that f is also an edge-magic labeling of G with valence val(f ) = 3(p+q +1)−val(f ). In the case of a super edge-magic labeling f of a graph G, there is also the corresponding super edge-magic complementary labeling, f c , which is also super edge-magic. In this case f c is defined by the rule f c (x) = p+1−f (x), ∀x ∈ V (G) and f c (ab) is obtained as described in Lemma 1.1, for all ab ∈ E(G). Then, the valence of f c can be expressed in terms of the valence of f as follows: The complementary labeling of an edge-magic labeling is a powerful tool that allows us to increase the number of valences of certain families of graphs dramatically. Using the complementary labeling we may even prove the perfect edge-magicness of many graphs. The following proposition can serve as an illustration of this fact.
Proposition 2.1. The graph K l 1,n is perfect edge-magic for all positive integers. Furthermore, Proof. Easy calculations show that J K l Also, in the case that G is a graph of equal order and size, new edge-magic labelings can be obtained from known super edge-magic labelings of G. The odd labeling and the even labeling [13] obtained from f , denoted respectively by o(f ) and e(f ), are the labelings o(f ), e(f ) : At this point, we want to observe that Proposition 2.1 can also be proved using the labelings provided in the proof of Lemma 1.3 and the odd and even labelings just defined above.
In [6], Figueroa et al. defined the following product: 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. a, b)). Note that when h is constant, D ⊗ h Γ is the Kronecker product. 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, [9,11,14,15]). 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 edge-magic digraph F is in S k p if |V (F )| = |E(F )| = p and the minimum sum of the labels of the adjacent vertices is equal to k (see Lemma 1.1). Notice that, since each 1-regular digraph has minimum edge induced sum equal to (p + 3)/2, it follows that S p ⊂ S (p+3)/2 p . The following result was introduced in [14], generalizing a previous result found in [6] : Let D be a (super) edge-magic digraph and let h : Remark 2.1. The key point in the proof of Theorem 2.1 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: , 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 Thus, the next result is obtained.
Letf be the (super) edge-magic labeling of the graph D ⊗ h S k p induced by a (super) edge-magic labeling f of D (see Remark 2.1). Then the valence off is given by the formula To prove the main result, we need some technical lemmas. The next lemma was proved in [12].
The following lemma was partially proved in [12].
Corollary 2.1. Let p and q be different odd primes. Then, there exist exactly 2p k−1 integers y with 1 < y < p k q such that gcd(y, p k q) = 1, gcd(y − 1, p k q) = 1. Moreover, these integers are of the form x + λpq, x ′ + λpq ∈ [1, p k q], where λ is a integer in [0, p k−1 − 1] and x, x ′ are the numbers described in Lemma 2.4.

3.
A family of perfect edge-magic graphs of the form C m ⊙ K n Let L be the set of vertices of degree 1 of G = C m ⊙ K n and C = V (G) \ L. Assume that ..,n i=0,1,2,...,m−1 where + m denotes the sum modulo m. Let − → G be an orientation of G such that, the subdigraph induced by C is strongly connected and all vertices of degree 1 have indegree 1. Note that, ,n is the digraph obtained by orienting K l 1,n in such a way that all vertices of degree 1 have indegree 1 and C + m is a strong orientation of C m . The following construction and lemmas are inspirated by the construction introduced by López et al. in [12]. Let M m be the set of all matrices of order m × m and let g 1 be the labeling of − → G induced by the product − → K l 1,n ⊗ C + m , when considering the super edgemagic labeling of − → K l 1,n that assigns label 1 to the central vertex and a super edge-magic labeling g of C m . By identifying each vertex of − → G with the label assigned to it by g 1 , we can construct the adjacency matrix of the digraph − → G , which is of the form: ij ∈ M m , A 1 ij = 0 for i > 1 and A 1 1j has the structure of the adjacency matrix of g(C m ) + , in which each vertex of C + m is identified with the label assigned to it by g. We can also consider the opposite strong orientation of the labeled cycle denoted by g(C m ) − . If we identify each vertex of − → G ∼ = − → K l 1,n ⊗ C − m with the labels induced by the product, we obtain an adjacency matrix of − → G with the same structure as A 1 g . Let us denote this matrix by B 1 g . Then B 1 g = (B 1 ij ), where each B 1 ij ∈ M m , B 1 ij = 0 for i > 1 and B 1 1j has the structure of the adjacency matrix of g(C m ) − , in which each vertex of C − m is identified with the label assigned to it by g.
Let A r g and B r g be the matrices obtained from A 1 g and B 1 g respectively by translating each row r − 1 units, for 1 ≤ r ≤ mn + 1. Thus, if A r g = (a r ij ), then Let G(A r g ) and G(B r g ) be the digraphs with adjacency matrices A r g and B r g respectively. We also denote by S(A r g ) and S(B r g ) the subdigraphs of G(A r g ) and G(B r g ) induced by the set of vertices {r, . . . , r − 1 + m}, respectively. From the adjacency matrices A r g and B r g , it is easy to check the following lemma.
Lemma 3.1. Let g be a super edge-magic labeling of C m . The vertices of G(A r g ) and G(B r g ) define a super edge-magic labeling g + r and g − r , respectively, with valence val(g + r ) = val(g − r ) = val(g 1 ) + r − 1, 1 ≤ r ≤ mn + 1.
The digraphs S(A r g ) and S(B r g ) are 1-regular and the graphs und(G(A r g )) and und(G(B r g )) are of the form H r g ⊙ K n where H r g is a 2-regular graph. Moreover, H r g ∼ = H r+λm g , for every positive integer λ with r + λm ≤ (m + 1)n.
Proof. The first part of the lemma comes from Lemma 1.2, since the minimum induced sum of two adjacent vertices increases by one unit at every step of the translation, and for r = 1 this minimum sum is the minimum sum of adjacent vertices of a super edge-magic labeled cycle. The second part is due to the structure of the adjacency matrices.  Let f be the canonical labeling of the cycle. By Lemmas 3.1 and 3.2, we obtain that for all r with 1 ≤ r ≤ mn+1, with either gcd((m+1)/2−(r−1), m) = 1 or gcd((m−1)/2−(r−1), m) = 1, either A r f or B r f is the adjacency matrix of a super edge-magic labeled digraph, whose underlying graph is G. Moreover, if f r is the induced super edge-magic labeling of G, then val(f r ) = val(f 1 ) + r − 1. Notice that, by Lemma 2.2, val(f 1 ) = (5m + 3)/2 + 2mn. Now, we provide a construction to cover the missing valences of G. That is, val(f 1 ) + r − 1, with gcd((m + 1)/2 − (r − 1), m) = 1 and gcd(m − 1)/2 − (r − 1), m) = 1. What happens for this values is that, by Lemma 3.2, we can not guarantee that H r f is a cycle. Also, by Lemma 3.2, we have that H r f ∼ = H r+λm f , for every positive integer λ with r + λm ≤ (m + 1)n. Thus, in what follows, we will assume that n = 1.
Notice that, if we prove the existence of a super edge-magic labeling g r of G with valence val(f 1 ) + r − 1, then the other missing valence, namely, val(f 1 ) + (pq − 1)/2 + αp will be realized by the super edge-magic complementary labeling of g r , namely g c r (see (2.1)). Let g be the labeling of C + m induced by the product f (C q ) + ⊗ f (C p ) − , when considering the canonical super edge-magic labeling of C q and C p , respectively. We will prove that H r g ∼ = und(S(A r g )) is a cycle of length pq.
The magic interval of crowns of the form C m ⊙ K n was obtained in [13].

Lemma 3.3. [13]
Let m and n be positive integers with m ≥ 3. Then, the magic interval of C m ⊙ K n is given by Theorem 3.1 implies that for every element k included in the super edge-magic interval, there exists a super edge-magic labeling with valence k. Taking the complementary labeling of these labelings, we get that all natural numbers from 3mn + (3 + 7m)/2 up to 4mn + (3 + 7m)/2 appear as valences of edge-magic labelings of C m ⊙ K n . Therefore, in order to prove that C m ⊙ K n is perfect edge-magic, we only need to show that for each k ∈ N, with 3mn + (3 + 5m)/2 < k < 3mn + (3 + 7m)/2, there exists an edge-magic labeling with valence k. We do this using the odd and even labelings of the labelings f r and g r introduced in the proof of Theorem 3.1.
Lemma 3.4. Let m be the product of two different odd primes and let n be any positive integer. Then, for each k with 2mn + 3m + 1 ≤ k ≤ 4mn + 3m + 2 there exists an edge-magic labeling of C m ⊙ K n with valence k.

Edge-magic labelings of crowns
The fact that even cycles admit edge-magic labelings has been known for several decades already. See the next theorem.
In fact this result has been improved recently as shown in the next theorem. It is also worth to mention that McQuillian [16] has made important contributions in this direction.
Theorem 4.2. [15] Let m = 2 α p α 1 1 p α 2 2 . . . p α k k be the unique prime factorization (up to ordering) of an even number m. Then C m admits at least Σ k i=1 α i edge-magic labelings with at least Σ k i=1 α i mutually different magic sums. If α ≥ 2, this lower bound can be improved to Similarly, the next result was established in [15] for cycles of odd order.

Theorem 4.3. [15]
Let m = p α 1 1 p α 2 2 . . . p α k k be the unique prime factorization (up to ordering) of an odd number m. Then C m admits at least 1 + Σ k i=1 α i edge-magic labelings with at least 1 + Σ k i=1 α i mutually different magic sums.
Proof. Note that G ∼ = und(C + m ⊗ − → K l 1,n ). Let g and g be edge-magic labelings of C + m ⊗ − → K l 1,n and C + m respectively and let γ r be a super edge-magic labeling of − → K l 1,n that assigns label r to the central vertex with val(γ r ) = r + 2n + 3, 1 ≤ r ≤ n + 1. By Lemma 4.1, we get val( g r ) = (n + 1)[val(g) − 2] + r + 1. Thus, val( g) depends on the valences of g and r. We know that by Lemma 1.3, − → K l 1,n has n + 1 valences and by Theorem 4.2, C m has at least Σ k i=1 α i mutually different valences. Thus, using Lemma 4.1, G = C m ⊙ K n admits at least (Σ k i=1 α i )(n + 1) mutually different magic sums. If α ≥ 2, this lower bound can be improved to (1 + Σ k i=1 α i )(n + 1).
Similarly, using Theorem 4.3 and Lemmas 1.3 and 4.1, we can prove the next theorem.
Theorem 4.5. Let m = p α 1 1 p α 2 2 . . . p α k k be the unique prime factorization (up to ordering) of an odd number m. Then G = C m ⊙ K n admits at least (1 + Σ k i=1 α i )(n + 1) mutually different magic sums.
Let f be the canonical labeling of the cycle C p k q , where p and q are different odd primes and k is a positive integer. The construction provide in Section 3 guarantees the existence of a super edge-magic labeling of the crown C p k q ⊙K n , with valence val(f 1 ) + r − 1, for many values of r. The possible exceptions can be obtained from Corollary 2.1.
Open question 1. Prove or disprove that C p k q ⊙ K n , where p and q are different odd primes and k is a positive integer is perfect (super) edge-magic.