Orientable Z_n-distance Magic Labeling of the Cartesian Product of Many Cycles

The following generalization of distance magic graphs was introduced in [2]. A directed Z_n-distance magic labeling of an oriented graph $\overrightarrow{G}=(V,A)$ of order n is a bijection $\overrightarrow{\ell}\colon V \rightarrow Z_n$ with the property that there is a $\mu \in Z_n$ (called the magic constant) such that w(x)= \sum_{y\in N_{G}^{+}(x)} \overrightarrow{\ell}(y) - \sum_{y\in N_{G}^{-}(x)} \overrightarrow{\ell}(y)= \mu$ for every x \in V(G). If for a graph G there exists an orientation $\overrightarrow{G}$ such that there is a directed Z_n-distance magic labeling $\overrightarrow{\ell}$ for $\overrightarrow{G}$, we say that G is orientable Z_n-distance magic and the directed Z_n-distance magic labeling $\overrightarrow{\ell}$ we call an orientable Z_n-distance magic labeling. In this paper, we find orientable Z_n-distance magic labelings of the Cartesian product of cycles. In addition, we show that even-ordered hypercubes are orientable Z_n-distance magic.


Introduction
A distance magic labeling of an undirected graph G of order n is a bijection l : V (G) → {1, 2, ..., n} such that y∈N (x) l(y) is the same for every x ∈ V (G). For a comprehensive survey of distance magic labeling, we refer the reader to [1].
An orientable Z n -distance magic labeling of a graph, first introduced by Cichacz et al. in [2], is a generalization of distance magic labeling. Let G = (V, E) be an undirected graph on n vertices. Assigning a direction to the edges of G gives an oriented graph − → G = (V, A). We will use the notation − → xy to denote an edge directed from vertex x to vertex y. That is, the tail of the arc is x and the head is y. For a vertex x, the set of head endpoints adjacent to x is denoted by N − (x), while the set of tail endpoints is denoted by N + (x). A directed Z n -distance magic labeling of an oriented graph − → G (V, A) of order n is a bijection − → ℓ : V → Z n with the property that there is a µ ∈ Z n (called the magic constant) such that If for a graph G there exists an orientation − → G such that there is a directed Z n -distance magic labeling − → ℓ for − → G , we say that G is orientable Z n -distance magic and the directed distance magic labeling ℓ we call an orientable Z n -distance magic labeling.
Throughout this paper we will use the notation [n] to represent the set {0, 1, ..., n − 1} for a natural number n. Furthermore, for a given i ∈ [n] and any integer j, let i + j denote the smallest integer in [n] such that i + j ≡ i + j (mod n). For a set of integers S and a number a, let S + a = {s + a : s ∈ S}.
Let C n = {x 0 , x 1 , ..., x n−1 , x 0 } denote a cycle of length n. For the sake of orienting the cycle, if the edges are oriented such that every arc has the form − −−− → x i x i+1 for all i ∈ [n], then we say the cycle is oriented clockwise. On the other hand, if all the edges of the cycle are oriented such that every arc has the form − −−− → x i x i−1 for all i ∈ [n], then we say the cycle is oriented counter-clockwise.

Cartesian product of two cycles
The Cartesian product G✷H is a graph with the vertex set V (G) × V (H). Two vertices (g, h) and (g ′ , h ′ ) are adjacent in G✷H if and only if g = g ′ and h is adjacent with h ′ in H, or h = h ′ and g is adjacent with g ′ in G. Hypercubes are interesting graphs which arise via the Cartesian product of cycles. The hypercube of order 2k, Q 2k is equivalent to the graph C 4 C 4 ... C 4 , where C 4 appears k times in the product. This graph is 2k regular on 4 k vertices. Labeling hypercubes has provided the motivation for the following theorems. Recall the following theorem proved in [2].
The Cartesian product of cycles, C m C n is orientable Z mn -distance magic for all m ≥ 3 and n ≥ 3.
The next theorem lays the groundwork for labeling hypercubes.
Since the graph G H can be edge-decomposed into cycles of those two forms, we have oriented every edge in each copy of G H.
where the arithmetic is done modulo pm 2 .
Expressing l( k x j i ) in the following alternative way , makes it easy to see that l is clearly bijective. Then for any given k and for all i, j ∈ [m] we have 3. Cartesian product of many cycles Theorem 3.1. For any m ≥ 3, the Cartesian product C m C m ... C m is orientable Z m ndistance magic.
Proof. Let G n ∼ = C m C m ... C m , the Cartesian product of n C m 's. Then for n ≥ 2 we may describe G n recursively as G n ∼ = G n−1 C m . We also have that, |V (G n )| = m n , so the labeling will take place in Z m n . The proof is by induction. For n = 1, we apply the labeling x 0 0 , x 1 0 , ..., x m−1 0 −→ {0, 1, ..., m − 1} and orient the cycle clockwise. Clearly for j ∈ {0, m−1}, w(x j 0 ) = 2−m ≡ 2 (mod m) and for 0 < j < m−1 , we have w(x j 0 ) = 2 ≡ 2 (mod m), so G 1 is orientable distance magic. For n = 2, Theorem 2.2 gives that G 2 is orientable distance magic and using the nomenclature from Theorem 2.2, w( . Furthermore, for each fixed j, the labels of x j i belong to the set [m] + jm for both base cases. Now construct G n ∼ = G n−1 C m as follows.
. Let x j i denote an arbitrary vertex in the subgraph H j i . Then for any integers a, b let x j+b i+a denote the corresponding vertex in the isomorphic subgraph H j+b i+a . Using this terminology, we have N Gn Now assume G n−1 is orientable Z m n−1 -distance magic with labeling l ′ : V (G n−1 ) → Z m n−1 . Then in G n , apply l ′ and its corresponding orientation to H 0 ∼ = G n−1 . As in the base cases, we may assume that the labels of H j 0 belong to P j for j ∈ [m] and Next, orient all the edges in each subgraph H 1 , H 2 , ..., H m−1 as in H 0 . Then the only edges left to orient in G n are cycles of the type x j 0 , x j 1 , ..., x j m−1 for fixed j. Orient each of these cycles clockwise. Now define l : V (G n ) → Z m n as follows.
To show that l is a bijection, it suffices to show that for each fixed i, l : P j −→ P j−i + im n−1 for all j, since for each fixed i, j − i runs through [m] as j runs through [m]. Since the labels of H j 0 belong to P j for j ∈ [m], we have Therefore, it is clear that for each i ∈ [m], the label set used on This completes the labeling and orientation of G n .
Observe that l( Therefore, and c = i(m − 1)m n−2 + m n−1 I{0 ≤ j − 1 ≤ i − 1} and I is the indicator function. Then we can write We will now show that I = −1 when j ≡ i or i − 1 (mod m) and I = 0 otherwise.
We have now fully determined the weight induced by the subgraph H i for each i ∈ [m]. We have, We are ready to determine the weight of each vertex. To this end we have w( and we recall that the arithmetic is to be performed in Z m n .
x 4 x 5 x 1 x 6 x 2 x 7 x 3 x 8 Figure 1. Q 3 Consequently, l(x 1 ), l(x 3 ), l(x 5 ) must all be odd. But N (x 4 ) = {x 1 , x 3 , x 5 }, so w(x 4 ) = µ is odd, a contradiction. Therefore, it cannot be the case that all three of l(x 2 ), l(x 4 ), l(x 6 ) are even. In fact, because the graph is vertex transitive, we have shown that no vertex may be adjacent to three even labeled vertices. So it must be the case that every vertex is adjacent to exactly one even-labeled vertex and two odd-labeled vertices. But this is impossible since there are an equal number of odd and even elements in Z 8 .
Case 2. The proof of the odd µ case is essentially the same as Case 1 and is left to the reader.
Hence, Q 3 is not orientable Z 8 -distance magic.
We conclude this section with the following conjecture.

Conclusion
We have proven that any number of disjoint copies of the Cartesian product of two cycles is orientable Z n -distance magic. We have also shown that the Cartesian product of any number of a given cycle is orientable Z n -distance magic, a result which encompasses even-ordered hypercubes. Finally, we have shown that the two smallest odd-ordered hypercubes are not orientable Z n -distance magic graphs, and we conjecture that no odd-ordered hypercube is orientable Z ndistance magic.