The Jumping Knight and Other (Super) Edge-Magic Constructions

Let G be a graph of order p and size q with loops allowed. A bijective function f:V(G)∪E(G)→{i}i=1p+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f:V(G)\cup E(G)\rightarrow \{i\}_{i=1}^{p+q}}$$\end{document} is an edge-magic labeling of G if the sum f(u)+f(uv)+f(v)=k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(u)+f(uv)+f(v)=k}$$\end{document} is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that f(V(G))={i}i=1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(V(G))=\{i\}_{i=1}^{p}}$$\end{document} then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which are related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product to get edge-magic labelings of some families of graphs.


Introduction
For the undefined concepts and notation used in this paper, we refer the reader to either one the following sources [4,5,11,19]. However, 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. We also point out that the graphs and digraphs used in this paper may contain loops. Whenever we refer to graphs without loops, we will call them simple graphs. Let C n be the cycle of order n. We denote by C + n and by C − n the two possible strong orientations of the cycle C n , and we use the expression − → G to denote an oriented graph obtained from a graph G. We also denote by C 1 a loop graph, that is a graph of order 1 and size 1. A digon formed by two different vertices x and y of a digraph is a directed cycle with set of arcs {(x, y), (y, x)}.
Among permutations, the ones arising from perfectly shuffling a deck of cards have received special attention. Let us define our deck of cards in such a way that each card receives a number from 1 up to n, the number of cards in the deck, the card numbered 1 lying at the bottom of the deck, the card numbered 2 lying on top of the card numbered 1 and so on until we reach the card numbered n. We will call this ordering of the deck α n and this will be our initial ordering (the standard word in terms of [3]).
Next, we will describe three possible ways to perfectly shuffle α n , and we will denote, following Asveld's notation in [3], these three ways by S • , S andS. The first perfect shuffle operation, S • , called the original perfect shuffle operation, consists on cutting a deck of an even number of cards n into two equal parts, and then interleaving these two parts. Hence, applying this operation to the initial ordering α n , n even, we obtain: S • (α n ) = 1, k, 2, k + 1, 3, k + 2, . . . , where k = (n + 1)/2 . It is clear that S • (α n ) can be obtained applying a permutation π(S • ) to α n , however π(S • ) always fixes 1. To avoid this fixed point, we can slightly modify the operation S • , by changing the way in which we are interleaving the cards. We apply the operation S on α n such that the resulting order is, when n is even S(α n ) = k, 1, k + 1, 2, k + 2, 3, . . . , where k = (n + 1)/2 . When n is odd, we isolate n and put it on top of the shuffled deck. Once again, S(α n ) can be obtained by applying a permutation π(S) on α n , and in this case, it is not obvious to decide for which even n, π(S) has fixed points. Furthermore, it is not clear when the cycle representation of π(S) consists of a unique oriented cycle, or it is the vertex disjoint union of oriented cycles.
An operation on α n , that we will pay special attention to, isS. This reordering operation, in a way, can be thought as the dual of S, and hence, it is known as the dual shuffle operation: . . , 1, k, n, when n odd, k − 1, n, k − 2, n − 1, . . . , 1, k, when n even, where k = (n + 1)/2 . In this case, it is clear that π(S) can be defined according to the parity of n as follows: if n is even then π(S)(m) ≡ −2m (mod n + 1), and if n is odd, then π(S)(m) ≡ −2m (mod n), for 1 ≤ m < n, and π(S)(n) = n. When the cycle representation associated to the dual shuffle operation on α n consists of a unique cycle, we say that n is a dual shuffle prime. An interesting question is to find an explicit formula for the nth-dual shuffle prime. We list below the first elements of the set of dual shuffle primes, namely, DSP. A more complete list can be found in [18]. Super edge-magic labelings are of importance among graph labelings due to the great amount of relations that they have with other labelings (see [7,10,13,15,16]). In particular, super edge-magic labelings of 2-regular graphs have proven to be of a great help to find links among labelings. Motivated by this fact, one of the goals of this paper is to use operations on α n to be able to find super edge-magic labelings of 2-regular graphs and other related graphs. 220 S. C. López et al. MJOM

Two Special Partitions
In this paper, two special partitions of the set {1, 2, . . . , n} emerge when studying the graphs obtained by our constructions. One type is of the form ∪ n k=1 Θ k , when n is odd, and the other one is of the form ∪ n k=1 Φ k , when n is even. Those sets, Θ k and Φ k are introduced just before Theorems 2.9 and 3.7, respectively, in a purely arithmetic manner.

Digraph Products Applied to Labelings
Figueroa-Centeno et al. introduced the following product of digraphs in [10]: let D be a digraph and let Γ = Let S n denote the set of all 1-regular super edge-magic labeled digraphs of order n where each vertex takes the name of the label assigned to it. The following results were also introduced in [10]: Theorem 1.2 [10]. Assume that D is any (super) edge-magic digraph and let h : E(D) −→ S n be any function. Then und(D ⊗ h S n ) is (super) edge-magic. Theorem 1.3 [10]. Let h be a function that assigns to each of the arcs of the digraph C + m the same strong orientation of the cycle C n . Then und(C + m ⊗ h S n ) ∼ = gcd(m, n) C lcm(m,n) . Theorem 1.2 has been extended by replacing the set S n by the set S k n in [16]. A super edge-magic labeled digraph F is in S k n if |V (F )| = |E(F )| = n and the minimum sum of the labels of the adjacent vertices is equal to k. An easy computation shows that S n ⊂ S (n+3)/2 n and, therefore, the next result is a generalization of Theorem 1.2. Theorem 1.4 [16]. Assume that D is any (super) edge-magic digraph and let h : E(D) −→ S k n be any function. Then und(D ⊗ h S k n ) is (super) edge-magic. As the title of the paper indicates we will devote these pages to construct edge-magic labelings of some infinite families of graphs. One of the main tools will be the ⊗ h -product. In what follows, we introduce the necessary notation and terminology as well as a brief survey of what is known about these families.
Let G and H be two (di)graphs, and let u be a distinguished vertex of G. Let H * uG be the (di)graph obtained from H and G by gluing a copy of G to each vertex x of H, by means of identifying x with the vertex playing the role of u in the given copy. Note that, in many circumstances, the resulting (di)graph H * uG is independent of the choice of the vertex u in G. In Sects. 4 Vol. 11 (2014) The Jumping Knight and Other (Super) Edge-Magic 221 and 5, we concentrate our attention on the cases when H and G are both cycles and when H is a cycle and G is a path.
Next, we introduce the following related results.
Theorem 1.5 [16]. Let m and n be positive odd integers with m ≡ 3 (mod 4) and n ≥ 3. If m ≥ n then C n * uC m is super edge-magic, u ∈ V (C m ).
Let P 2n+1 denote the path of order 2n + 1. We have the following result.
Theorem 1.6 [16]. If C m is (super) edge-magic and v is the central vertex of The following open question can be found in [16].
Question 1.1 [16]. Characterize the pairs (n, m) for which C n * uC m , u ∈ V (C m ), is super edge-magic.
In this paper, we introduce the following open question and we devote Sect. 4 to throw some light towards a possible solution.
The next results can be found in [10] and [2]. Theorem 1.7 [10]. Let G be any bipartite graph with stable sets V 1 and V 2 , and let − → G be the digraph obtained from G by orienting each edge of Theorem 1.8 [10]. Let − → T be any orientation of a tree T , and let h : Theorem 1.9 [2]. Let m, n ∈ N and consider the product The organization of the paper is the following one. We construct super edge-magic labelings of 2-regular graphs using the classical jump of the knight on the chess game in Sect. 2. With a slight modification, we obtain super edge-magic labelings of the union of 2-regular graphs with a K 2 in Sect. 3. In Sects. 4 and 5, we study the edge-magicness of C n * uC m and C n * uP m , respectively, using the ⊗ h -product of digraphs. Some further results obtained from the previous sections are given in Sect. 6. A more detailed summary of the results presented in the paper is given in the final Sect. 6.1.

Super Edge-Magic Labelings of 2-Regular Graphs
It is well known that if a (p, q)-graph G is super edge-magic then q ≤ 2p − 3 (see [6]). If G is bipartite, then this bound can be improved to q ≤ 2p − 5 (see [8]). These bounds imply that if an r-regular graph is super edgemagic, then r ∈ {0, 1, 2, 3}. It is clear that 0-regular graphs are of no interest in this context. Furthermore, 1-regular super edge-magic graphs have been completely characterized in [14] by Kotzig and Rosa. Therefore, it seems to be a natural question to ask which 2-regular graphs are super edge-magic. This is what was in the minds of Figueroa-Centeno et al. in [9] when they conjectured that the graph C m ∪ C n is super edge-magic if and only if m + n is odd and greater than 1. Holden et al. went further into this conjecture, although they arrived to it from a different point of view, when they conjectured in [12] that all 2-regular graphs of odd order are strong vertex total magic, excluding C 3 ∪ C 4 , 3C 3 ∪ C 4 and 2C 3 ∪ C 5 , which is in fact equivalent to saying that they are super edge-magic.
In this section, we establish a new relationship between super edgemagic labelings of 2-regular graphs and π(S). Let D λ be a super edge-magic digraph D with a super edge-magic labeling λ, such that each vertex of D takes the name of its label under λ. Then, the adjacency matrix of D λ , we let the diagonals from left to right and bottom to top be the counterdiagonals, then the set of counterdiagonals containing a 1 is a consecutive set. Furthermore, each counterdiagonal contains at most a 1. If D λ is a digraph obtained from a 2-regular graph by attaching a cyclic orientation to each component, that is to say, D λ is a 1-regular digraph, then A(D λ ) has the extra property that each column and each row contains exactly a 1. Moreover, it is clear that if a square matrix A of 0's and 1's meets the following properties: (i) each counterdiagonal contains at most one 1, (ii) the counterdiagonals containing exactly a 1 form a set of consecutive counterdiagonals, and (iii) each row and each column of the matrix have all entries being 0, except for exactly one entry which is 1, then A is the adjacency matrix of some super edge-magic labeled 1-regular digraph.

The Jumping Knight
Let A n = (a ij ) be the square matrix of dimension n defined by a ij = 1 if j ≡ −2i + 2 (mod n) and a ij = 0, otherwise. Similarly, let A n = (a ij ) be the square matrix of dimension n defined by a ij = 1 if j ≡ −2i + 1 (mod n) and a ij = 0, otherwise. Observe that the positions of the 1 entries of the matrices A n and A n so defined, correspond to the jump of the knight on the n × n chessboard, when the matrices are viewed as chessboards with exactly n 2 squares.
Let G n and G n be the digraphs with vertex sets V (G n ) = V (G n ) = {1, 2, . . . , n} and adjacency matrices A n and A n , respectively. The function Lemma 2.1. Let n be an odd integer. Then the digraph G n is a 1-regular super edge-magic digraph.
Proof. It is clear since its adjacency matrix meets the three properties mentioned above.
Proof. Let x ∈ {1, 2, . . . , n} and consider the sequence The next lemma introduces a nice integer sequence that will be used to study the cycle structure of the digraph G n . Proof. Equality (i) is an easy exercise. Since a 3 = 3, condition (ii) clearly holds for l = 1. Suppose that it holds for 1 ≤ i ≤ l and consider the equalities Thus, we obtain that a 3 l+1 /a 3 l = (a 3 l ) 2 − 3 · 2 3 l . Hence, 3 is a divisor of a 3 l+1 /a 3 l and 9 is not. Therefore, and by the induction hypothesis, we obtain that 3 l+1 is a divisor of a 3 l+1 and 3 l+2 is not.
By replacing z by −2 in the polynomial factorization (2.1) Remark 2.4. The integer sequence introduced in Lemma 2.3 is called the Jacobsthal sequence (or Jacobsthal numbers). This sequence, which appears in [18] as 'A001045', has connections with multiple applications, some of them can be found in [18]. The next result shows another one. MJOM where a 3 s = 3 s m and gcd(3, m) = 1. Thus, by multiplying by 3 k−1−s n 1 , the congruence 0 ≡ 2 · 3 k−1 n 1 m (mod n) follows, a contradiction. Let us see now that C 3 k is a subgraph of und(G n ). Let x be a solution of 3x ≡ 2 (mod n 1 ). Thus, we obtain that 3 k+1 mx ≡ 2 · 3 k m (mod 3 k n 1 ), where a 3 k = 3 k m. Hence, by Corollary 2.5, there is a cycle of length 3 k . This proves (ii). Finally, let us see (iii). Let n = 3 k . By using a similar proof as in (ii), it is clear that und(G n ) does not contain a cycle of length 3 s , for 1 ≤ s < k. Moreover, each x ∈ {1, 2, . . . , n} is a solution of the trivial equation 3a 3 k x ≡ 2 · a 3 k (mod n), since a 3 k ≡ 0 (mod n).
Notice that from the previous proof, we also conclude that G n contains a cycle of length 3 k , for some integer k ≥ 0, if and only if 3 k is a divisor of n and 3 k+1 is not. The characterization of the cycle structure of G n is provided in the next three results.

Proposition 2.7. Let n be an odd prime. Let k be the minimum i such that
Proof. Consider the congruence relation obtained in Corollary 2.5, 3a i x ≡ 2a i (mod n). Hence, since n is prime, either a i ≡ 0 (mod n), or 3x ≡ 2 (mod n). That is, either (−2) i ≡ 1 (mod n), or 3x ≡ 2 (mod n). Therefore, und(G n ) ∼ = (n − 1)/kC k ∪ C 1 , where C 1 is the graph defined by a loop in the only x ∈ {1, 2, . . . , n} such that 3x ≡ 2 (mod n). Proof. By Theorem 2.11 in [3], n is a dual shuffle prime if and only if, n + 1 is prime and −2 generates the multiplicative group Z * n+1 . Thus, by Proposition 2.7, this is the case, if and only if, und(G n+1 ) ∼ = C n ∪ C 1 .
Let n be an odd integer, (a i ) the Jacobsthal sequence and let Θ k be a subset of {1, 2, . . . , n} defined by x ∈ Θ k if k is the minimum i with 3a i x ≡ 2a i (mod n). Then, from Corollary 2.5, it is easy to obtain the following result. Theorem 2.9. Let n be an odd integer. Then, Theorem 2.9 and Lemma 2.1 imply the next result.

Theorem 2.10. Let n be an odd integer. Then, the graph ∪ n k=1 |(Θ k |/k)C k is super edge-magic.
Notice that, by Theorem 2.9, we immediately obtain that the sets Θ k , for 1 ≤ k ≤ n, partition the set {1, 2, . . . , n}, when n is odd. However, this does not hold when n is even.

Super Edge-Magic Labelings of 2-Regular Graphs Union K 2
The construction shown in Sect. 2.1 can be slightly modified to give super edge-magic labelings of the union of K 2 and 2-regular graphs, we feel that this is interesting since Wallis proposes in [19] the following open question that appears in his book as research problem 2.15. Question 3.1 [19]. For which values of n is C n ∪ K 2 edge-magic?
Using this technique, we are able to find values of n for which C n ∪ K 2 is super edge-magic, and therefore edge-magic.

Proposition 3.3.
Let n be an even integer such that n + 1 is prime. Let k be the minimum i such that (−2) i ≡ 1 (mod n + 1). Then, Proof. Let k be the minimum i such that (−2) i ≡ 1 (mod n + 1). By Lemma 3.2, we know that und(J n ) contains a component isomorphic to a cycle of order k as a subgraph. Furthermore, 1 is in the vertex set of this component. Suppose that x ∈ {1, 2, . . . , n} belongs to a component of und(J n ) isomorphic to a cycle of order r, then r is the minimum i such that ((−2) i − 1)x ≡ 0 (mod n + 1). Thus, since n + 1 is prime, we have that r is the minimum i such that (−2) i − 1 ≡ 0 (mod n + 1). That is, by hypothesis r = k. Proof. By Theorem 2.11 in [3], n is a dual shuffle prime if and only if, −2 generates the multiplicative group Z * n+1 . Thus, by Proposition 3.3, n is a dual shuffle prime if and only if, und(H n+1 ) ∼ = C n ∪ K 2 . Lemma 3.5. Let n be an even integer. If n ≡ 0 (mod 4) then C n ∪ K 2 is super edge-magic.
defined by, f (x) = 1, f (y) = n + 2 and: Then, f is a super edge-magic labeling of C n ∪ K 2 .
The next corollary is an easy consequence of the previous lemma together with Corollary 3.4 and Lemma 3.1. It is worth mentioning, that Park et al. gave a super edge-magic labeling of C n ∪ K 2 , for n even, n = 10, in [17]. However, their description was somewhat more complicated. Furthermore, the smallest and the largest labels in our labelings are assigned to the vertices of K 2 . Next, we introduce a result that is, in fact, easily deduced from Lemma 3.2.
Let n be an even integer and let Φ k ⊂ {1, 2, . . . , n} such that x ∈ Φ k if k is the minimum i with the property that ((−2) i − 1)x ≡ 0 (mod n + 1). Theorem 3.7. Let n be an even integer. Then, Finally, we have the following result that can be obtained from Theorem 3.7 and Lemma 3.1.

Theorem 3.8. Let n be an even integer. Then, the graph
Notice that, by Theorem 3.7, we immediately obtain that the sets Φ k , for 1 ≤ k ≤ n, partition the set {1, 2, . . . , n}, when n is even. However, this does not hold when n is odd. The following open question is motivated by Question 1.1 and by the work conducted in this section.  To prove this theorem, first of all we will state and prove the following two lemmas.
Proof. Let − → C b m be an orientation of C m in which the arrows go from one stable set of C m to the other one, when m is even. Notice that, by the definition of * , it is clear that . Moreover, by the definition of the ⊗ h -product, ). Hence, we only have to find a function h 1 Proof. As in the previous proof, we only need to find a function h 1 : Assume that m = n, otherwise the result holds by Theorem 1.3. Since n is odd, we have that the congruence relation can be solved for some r such that 0 < r < m. Therefore, inheriting the notation of Theorem 1.9, by considering any function h 1 with N 2 (h − 1 ) = r, we have that Theorem 1.9 implies that und( Now, we are ready to prove Theorem 4.1. Proof of Theorem 4.1. If m ≡ 3 (mod 4) and m ≥ n, by Theorem 1.5, we know that C n * uC m , u ∈ V (C m ) is super edge-magic, and hence edge-magic. In fact, using Theorem 1.2, Lemma 4.3 and the hypothesis on C u m we obtain that C n * uC m is (super) edge-magic for each odd integer m, with m ≥ n. Thus, we only need to concentrate on the case when m is even. This case follows from Theorem 1.2 and Lemma 4.2.
Consider the edge-magic labelings of C u 4 and C u 6 that appear in Fig. 1. Hence, we immediately obtain the following corollary of Theorem 4.1

The Edge-Magicness of C n * uP m
Let the graph P l m be the path of order m with a loop attached to one of the leaves, l, of P m . Then we introduce the following lemma.
Lemma 5.1. The graph P l m is edge-magic for every m ∈ N, where l is a leaf of P m .
Proof. We will consider two cases, depending on the parity of m.
i=1 defined by the rule: Then f is an edge-magic labeling of P l m . Case m is odd. For m odd, consider the function f : defined by the rule: Then f is an edge-magic labeling of P l m . At this point, we are ready to state and prove the following theorem.
Theorem 5.2. Let n ≥ 3 be an odd integer and consider the cycle C n . Also consider any path P m and let l be any leaf of P m . Then the graph C n * lP m is edge-magic.
Proof. Let − → P m be an orientation of P m that allows us to travel from one leaf of P m to the other one following the direction of the arrows. Let Γ = {C + n , C − n } and consider any function h : E( The result is trivial for P l 2 . Assume that m ≥ 6 and consider k ∈ N with m = 2(2k + 1), if i = 2k + 2, 3k − j + 2, if i = 2j, j = 1, . . . , k, 5k − j + 3, if i = 2j, j = k + 2, . . . , 2k + 1.
Then f is a super edge-magic labeling of P l m . Notice that the graph und( − → P l,s m ⊗ C + n ) is isomorphic to the graph obtained from P 2 × C n by replacing each edge joining two vertices of the different copies of C n by a path of order m. By Theorem 1.2 and Lemma 5.5 we obtain the next result.
Corollary 5.6. Let m be an even integer and n be odd. Then, the graph und( − → P l,s m ⊗ C + n ) is super edge-magic.

Further Results and Conclusions
Using similar techniques as the ones presented in Sects. 4 and 5, we can extend the family of super edge-magic graphs. In particular, we can extend the family of 2-regular super edge-magic graphs. For instance, when m is a dual shuffle prime we can obtain the following result, which is a particular case of a more general result found in [16].
Theorem 6.1. Let m be a dual shuffle prime and let n be an odd integer n ≥ 3.
Then C mn ∪ C n is super edge-magic.