The following generalization of distance magic graphs was introduced in [2]. A directed ℤn- distance magic labeling of an oriented graph G = (V,A) of order n is a bijection ℓ: V → ℤn with the property that there is a μ ∈ ℤn (called the magic constant) such that If for a graph G there exists an orientation G such that there is a directed ℤn-distance magic labeling ℓ for G, we say that G is orientable ℤn-distance magic and the directed ℤn-distance magic labeling ℓ we call an orientable ℤn-distance magic labeling. In this paper, we find orientable ℤn- distance magic labelings of the Cartesian product of cycles. In addition, we show that even-ordered hypercubes are orientable ℤn-distance magic

Freyberg, Bryan

Keranen, Melissa S.

English

Michigan Technological University

Michigan Technological University Digital Commons @ Michigan Tech Michigan Tech Publications 2017 Orientable ℤ < inf> n -distance magic labeling of the Cartesian product of many cycles Bryan Freyberg Southwest Minnesota State University Melissa S. Keranen Michigan Technological University, msjukuri@mtu.edu Follow this and additional works at: https://digitalcommons.mtu.edu/michigantech-p  Part of the Mathematics Commons Recommended Citation Freyberg, B., & Keranen, M. S. (2017). Orientable ℤ < inf> n -distance magic labeling of the Cartesian product of many cycles. Electronic Journal of Graph Theory and Applications, 5(2), 304-311. http://doi.org/10.5614/ejgta.2017.5.2.11 Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/3217 Follow this and additional works at: https://digitalcommons.mtu.edu/michigantech-p  Part of the Mathematics Commons brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Michigan Technological Universitywww.ejgta.orgElectronic Journal of Graph Theory and Applications 5 (2) (2017), 304–311Orientable Zn-distance magic labeling of theCartesian product of many cyclesBryan Freyberga, Melissa KeranenbaDepartment of Mathematics and Computer Science, Southwest Minnesota State University, Marshall, MN56258bDepartment of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931bryan.freyberg@smsu.edu, msjukuri@mtu.eduAbstractThe following generalization of distance magic graphs was introduced in [2]. A directed Zn-distance magic labeling of an oriented graph−→G = (V,A) of order n is a bijection−→` : V → Znwith the property that there is a µ ∈ Zn (called the magic constant) such thatw(x) =∑y∈N+G (x)−→` (y)−∑y∈N−G (x)−→` (y) = µ for every x ∈ V (G).If for a graph G there exists an orientation−→G such that there is a directed Zn-distance magic label-ing−→` for−→G , we say that G is orientable Zn-distance magic and the directed Zn-distance magiclabeling−→` we call an orientable Zn-distance magic labeling. In this paper, we find orientable Zn-distance magic labelings of the Cartesian product of cycles. In addition, we show that even-orderedhypercubes are orientable Zn-distance magic.Keywords: distance magic graph, directed distance magic labeling, orientable Zn-distance magic labelingMathematics Subject Classification : 05C78DOI: 10.5614/ejgta.2017.5.2.11Received: 20 September 2016, Revised: 30 July 2017, Accepted: 8 September 2017.304www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. Keranen1. IntroductionA 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 ofdistance magic labeling, we refer the reader to [1].An orientable Zn-distance magic labeling of a graph, first introduced by Cichacz et al. in [2], isa 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 thenotation −→xy to denote an edge directed from vertex x to vertex y. That is, the tail of the arc is x andthe head is y. For a vertex x, the set of head endpoints adjacent to x is denoted by N−(x), while theset of tail endpoints is denoted by N+(x). A directed Zn-distance magic labeling of an orientedgraph−→G(V,A) of order n is a bijection−→` : V → Zn with the property that there is a µ ∈ Zn(called the magic constant) such thatw(x) =∑y∈N+G (x)−→` (y)−∑y∈N−G (x)−→` (y) = µ for every x ∈ V (G).If for a graph G there exists an orientation−→G such that there is a directed Zn-distance magiclabeling−→` for−→G , we say that G is orientable Zn-distance magic and the directed distance magiclabeling ` we call an orientable Zn-distance magic labeling.Throughout this paper we will use the notation [n] to represent the set {0, 1, ..., n− 1} fora natural number n. Furthermore, for a given i ∈ [n] and any integer j, let i + j denote thesmallest integer in [n] such that i+ j ≡ i + j (modn). For a set of integers S and a number a, letS + a = {s+ a : s ∈ S}.Let Cn = {x0, x1, ..., xn−1, x0} 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 −−−−→xixi+1 for all i ∈ [n], then we say thecycle is oriented clockwise. On the other hand, if all the edges of the cycle are oriented such thatevery arc has the form −−−−→xixi−1 for all i ∈ [n], then we say the cycle is oriented counter-clockwise.2. Cartesian product of two cyclesThe Cartesian product G2H is a graph with the vertex set V (G)× V (H). Two vertices (g, h)and (g′, h′) are adjacent in G2H if and only if g = g′ and h is adjacent with h′ in H , or h = h′ andg is adjacent with g′ in G. Hypercubes are interesting graphs which arise via the Cartesian productof cycles. The hypercube of order 2k, Q2k is equivalent to the graph C4C4...C4, where C4appears k times in the product. This graph is 2k regular on 4k vertices. Labeling hypercubes hasprovided the motivation for the following theorems. Recall the following theorem proved in [2].Theorem 2.1. ([2]) The Cartesian product of cycles, CmCn is orientable Zmn-distance magicfor all m ≥ 3 and n ≥ 3.The next theorem lays the groundwork for labeling hypercubes.305www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. KeranenTheorem 2.2. For any p ≥ 1, and m ≥ 3, p disjoint copies of the graph CmCm is orientableZpm2-distance magic.Proof. Let G ∼= Cm ∼= {g0, g1, ..., gm−1, g0} ∼= {h0, h1, ..., hm−1, h0} ∼= H . Then orient each copyof GH as follows. Fix j ∈ [m]. Then for all i ∈ [m], orient clockwise each cycle of the form{(gi, hj) , (gi+1, hj) , ..., (gi−1, hj) , (gi, hj)}. Similarly, fix i ∈ [m]. Then for all j ∈ [m], orientclockwise each cycle of the form {(gi, hj) , (gi, hj+1) , ..., (gi, hj−1) , (gi, hj)}. Since the graphGH can be edge-decomposed into cycles of those two forms, we have oriented every edge ineach copy of GH . Now let kxji denote the vertex (gi, hj) of the kth copy of GH for i, j ∈ [m],and k ∈ {1, 2, ..., p}. Then, for each k ∈ {1, 2, ..., p}, define−→l : V → Zpm2 by~l(kxji ) = pmj + (k − 1)m+ i − j,where the arithmetic is done modulo pm2.Expressing ~l(kxji ) in the following alternative way~l(kxji ) =pmj + (k − 1)m,pmj + (k − 1)m+ 1,pmj + (k − 1)m+ 2,...pmj + (k − 1)m+ (m− 1),for i ≡ j (modm)for i ≡ j + 1 (modm)for i ≡ j + 2 (modm)...for i ≡ j − 1 (modm),makes it easy to see that ~l is clearly bijective. Then for any given k and for all i, j ∈ [m] we haveN+(kxji)= {kxj+1i , kxji+1} and N−(kxji)= {kxj−1i , kxji−1}. Recalling that w(kxji ) ∈ Zpm2 ,we havew(kxji ) =~l(kxj+1i ) +~l(kxji+1)− [~l(kxji−1) +~l(kxj−1i )]= [j + 1+ j − j − j + 1]pm+ (i − j − 1)− (i − j − 1) + (i − j + 1)− (i − j + 1)= [(j + 1)− (j − 1)]pm={(2−m)pm, j ∈ {0,m− 1}2pm, 0 < j < m− 1= 2pm,so~l is an orientable Zpm2-distance magic labeling. Hence, p copies of CmCm is orientable Zpm2-distance magic.3. Cartesian product of many cyclesTheorem 3.1. For any m ≥ 3, the Cartesian product CmCm...Cm is orientable Zmn-distance magic.306www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. KeranenProof. Let Gn ∼= CmCm...Cm, the Cartesian product of n Cm’s. Then for n ≥ 2 wemay describe Gn recursively as Gn ∼= Gn−1Cm. We also have that, |V (Gn)| = mn, so thelabeling will take place in Zmn . The proof is by induction. For n = 1, we apply the label-ing{x00, x10, ..., xm−10}7−→ {0, 1, ...,m− 1} and orient the cycle clockwise. Clearly for j ∈{0,m−1},w(xj0) = 2−m ≡ 2 (modm) and for 0 < j < m−1 , we havew(xj0) = 2 ≡ 2 (modm),so G1 is orientable distance magic. For n = 2, Theorem 2.2 gives that G2 is orientable distancemagic and using the nomenclature from Theorem 2.2, w(xji ) = (2 − m)m ≡ 2m (modm2) forj ∈ {0,m − 1}, i ∈ [m] and w(xji ) = 2m for j ∈ {0 < j < m − 1}, i ∈ [m]. Further-more, for each fixed j, the labels of xji belong to the set [m] + jm for both base cases. Nowconstruct Gn ∼= Gn−1Cm as follows. Let H i ∼= Gn−1 for i ∈ [m]. Furthermore, for a giveni, let Hji ∼= Gn−2 for j ∈ [m]. Let xji denote an arbitrary vertex in the subgraph Hji . Then forany integers a, b let xj+bi+a denote the corresponding vertex in the isomorphic subgraph Hj+bi+a . Us-ing this terminology, we have NGn(xji)= NGn−1(xji ) ∪{xji+1, xji−1}. Let wHi(xji ) denote theweight (in Zmn) induced on xji by the subgraph H i and wHji (xji ) denote the weight (again in Zmn)induced on xji by the subgraph Hji . Now partition Zmn−1 = P0 ∪ P1 ∪ P2 ∪ ... ∪ Pm−1 so thatPj = [mn−2] + jmn−2 for j ∈ [m].Now assume Gn−1 is orientable Zmn−1-distance magic with labeling ~l′ : V (Gn−1) → Zmn−1 .Then in Gn, apply ~l′ and its corresponding orientation to H0 ∼= Gn−1. As in the base cases, wemay assume that the labels of Hj0 belong to P j for j ∈ [m] andwH0(xj0) ={(2−m)mn−2, j ∈ {0,m− 1}2mn−2, j ∈ {1, 2, ...m− 2} .Next, orient all the edges in each subgraph H1, H2, ..., Hm−1 as in H0. Then the only edgesleft to orient in Gn are cycles of the type{xj0, xj1, ..., xjm−1}for fixed j. Orient each of these cyclesclockwise. Now define ~l : V (Gn)→ Zmn as follows.~l(xji ) ={~l′(xj0) + i(m− 1)mn−2 +mn−1, 0 ≤ j < i~l′(xj0) + i(m− 1)mn−2, i ≤ j ≤ m− 1.To show that ~l is a bijection, it suffices to show that for each fixed i, ~l : Pj 7−→ Pj−i + imn−1 forall j, since for each fixed i, j − i runs through [m] as j runs through [m]. Since the labels of Hj0belong to P j for j ∈ [m], we have~l : Pj 7−→{Pj + i(m− 1)mn−2 +mn−1, 0 ≤ j < iPj + i(m− 1)mn−2, i ≤ j ≤ m− 1.Now, if 0 ≤ j < i, we havePj 7→ Pj + i(m− 1)mn−2 +mn−1= [mn−2] + jmn−2 + i(m− 1)mn−2 +mn−1= [mn−2] + (j − i)mn−2 + (i+ 1)mn−1= [mn−2] + (j − i)mn−2 +mmn−2 + imn−1= [mn−2] + (m+ j − i)mn−2 + imn−1= Pj−i + imn−1.307www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. KeranenWhile if i ≤ j ≤ m− 1, we havePj 7→ Pj + i(m− 1)mn−2= [mn−2] + jmn−2 + i(m− 1)mn−2= [mn−2] + (j − i)mn−2 + imn−1= Pj−i + imn−1.Therefore, it is clear that for each i ∈ [m], the label set used on H i is i ·mn−1 + {P0 ∪ P1 ∪ ... ∪Pm−1} = Zmn−1 + imn−1, so we see that−→l : V (Gn) → Zmn is bijective. This completes thelabeling and orientation of Gn.Observe that ~l(xji ) ≡ ~l′(xj0) (modmn−2). Therefore, wHji (xji ) = wHj0(xj0) in Zmn . Then wehave,wHi(xji ) = wHji(xji ) +~l(xj+1i )−~l(xj−1i )= wHj0(xj0) +~l(xj+1i )−~l(xj−1i ).But,wHj0(xj0) = wH0(xj0)− [~l′(xj+10 )− ~l′(xj−10 )].Therefore,wHi(xji ) = wH0(xj0)− ~l′(xj+10 ) +~l′(xj−10 ) +~l(xj+1i )−~l(xj−1i )= wH0(xj0) + [~l(xj+1i )− ~l′(xj+10 )]− [(~l(xj−1i )− ~l′(xj−10 )]= a+ b− c,where a = 2mn−2−mn−1I{j = 0 or m− 1}, b = i(m− 1)mn−2+mn−1I{0 ≤ j + 1 ≤ i − 1},and c = i(m − 1)mn−2 +mn−1I{0 ≤ j − 1 ≤ i − 1} and I is the indicator function. Then wecan writewHi(xji ) = 2mn−2+mn−1 [I{0 ≤ j + 1 ≤ i − 1} − I{j = 0 or m− 1} − I{0 ≤ j − 1 ≤ i − 1}] .Let I = I{0 ≤ j + 1 ≤ i − 1}− I{j = 0 or m− 1}− I{0 ≤ j − 1 ≤ i − 1}. We will nowshow that I = −1 when j ≡ i or i− 1 (modm) and I = 0 otherwise.Case 1. Suppose j ≡ i (modm). Then since j, i ∈ [m], we have j = i and hence I = 1− 1− 1 =−1 when j = 0 or j = m− 1. When 1 ≤ j ≤ m− 2, we have I = 0− 0− 1 = −1.Case 2. Suppose j ≡ i− 1 (modm). Then if j = 0, we have I = 0− 1− 0 = −1. If j = m− 1,we have I = 1− 1− 1 = −1. If 1 ≤ j ≤ m− 2, we have I = 0− 0− 1 = −1.Case 3. Suppose j 6≡ i, i−1 (modm). Then if j = 0, we have I = 1−1−0 = 0. If j = m−1, wehave I = 1−1−0 = 0. If 1 ≤ j ≤ i − 2, we have I = 1−0−1 = 0. If i + 1 ≤ j ≤ m−2,we have I = 0− 0− 0 = 0, proving the claim.We have now fully determined the weight induced by the subgraph H i for each i ∈ [m]. We have,wHi(xji ) ={(2−m)mn−2, j ≡ i or i− 1 (modm)2mn−2, otherwise .We are ready to determine the weight of each vertex. To this end we have w(xji ) = wHi(xji ) +~l(xji+1)−~l(xji−1) and we recall that the arithmetic is to be performed in Zmn .308www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. KeranenCase 1. Suppose j ≡ i or i− 1 (modm). Then we havew(xji ) = (2−m)mn−2 +{2(m− 1)mn−2 +mn−1, 1 ≤ i ≤ m− 2(2−m)(m− 1)mn−2, i ∈ {0,m− 1}={2mn−1, 1 ≤ i ≤ m− 22mn−1 −mn, i ∈ {0,m− 1}≡ 2mn−1 (modmn),since (i + 1) − (i − 1) ≡ 2 (modmn) when 1 ≤ i ≤ m − 2 and (i + 1) − (i − 1) ≡2−m (modmn) when i ∈ {0,m− 1}.Case 2. Suppose j 6≡ i, i− 1 (modm). Then we havew(xji ) = 2mn−2 +{2(m− 1)mn−2, 1 ≤ i ≤ m− 2(2−m)(m− 1)mn−2 −mn−1, i ∈ {0,m− 1}={2mn−1, 1 ≤ i ≤ m− 22mn−1 −mn, i ∈ {0,m− 1}≡ 2mn−1 (modmn).Hence, w(xji ) = 2mn−1 for all i, j ∈ [m], so Gn is orientable Zmn-distance magic for all n ≥1.Corollary 3.1. The hypercube Q2k is orientable Z22k-distance magic for all k ≥ 1.Proof. Since Q2k ∼= C4C4...C4, the Cartesian product of k C4’s, Theorem 3.1 gives theresult.One may wonder if the hypercube Q2k+1 is orientable Z22k+1-distance magic. Since the graphis odd regular, a little pessimism is understandable. Indeed, no odd regular graph on n ≡ 2 (mod 4)vertices is orientable Zn-distance magic, as was proved in [2]. However,Q2k+1 contains 22k+1vertices,a number divisible by 4, so it is possible that Q2k+1 is orientable Z22k+1-distance magic for somek. It can easily be checked that Q1 ∼= K2 is not. The following theorem rules out Q3 as well.Theorem 3.2. The hypercube Q3 is not orientable Z8-distance magic.Proof. Let G ∼= Q3 as shown in Figure 1. An important fact we will use is that regardless ofthe orientation of the edges, the (directed) weight of a given vertex has the same parity as thesum (performed in Z8 of course) of its neighbors. Suppose for the sake of contradiction that Gis orientable Z8-distance magic with orientable Z8-distance magic labeling ~l : V (G) → Z8 andassociated magic constant µ.Case 1. Suppose µ is even. Observe that N(x1) = {x2, x4, x6}. Then since µ is even, either allthree of ~l(x2),~l(x4),~l(x6) are even or exactly one is even. Suppose it is the case that allthree of ~l(x2),~l(x4),~l(x6) are even. Then that leaves but one other vertex with an even label.Since N(x3) = {x2, x4, x8}, and w(x3) = µ is even, it must be the case that ~l(x8) is even.309www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. Keranenx4x5x1x6x2x7x3x8Figure 1. Q3Consequently, ~l(x1),~l(x3),~l(x5) must all be odd. But N(x4) = {x1, x3, x5}, so w(x4) = µis odd, a contradiction. Therefore, it cannot be the case that all three of~l(x2),~l(x4),~l(x6) areeven. In fact, because the graph is vertex transitive, we have shown that no vertex may beadjacent to three even labeled vertices. So it must be the case that every vertex is adjacentto exactly one even-labeled vertex and two odd-labeled vertices. But this is impossible sincethere are an equal number of odd and even elements in Z8.Case 2. The proof of the odd µ case is essentially the same as Case 1 and is left to the reader.Hence, Q3 is not orientable Z8-distance magic.We conclude this section with the following conjecture.Conjecture 1. The odd-ordered hypercube, Q2k+1 is not orientable Z22k+1-distance magic for anyk ∈ {0, 1, 2, ...}.4. ConclusionWe have proven that any number of disjoint copies of the Cartesian product of two cycles isorientable Zn-distance magic. We have also shown that the Cartesian product of any number ofa given cycle is orientable Zn-distance magic, a result which encompasses even-ordered hyper-cubes. Finally, we have shown that the two smallest odd-ordered hypercubes are not orientableZn-distance magic graphs, and we conjecture that no odd-ordered hypercube is orientable Zn-distance magic.AcknowledgementWe would like to thank Dalibor Froncek, University of Minnesota Duluth, for motivating The-orem 3.2 and for his encouragement in general.310www.ejgta.orgOrientable distance magic labeling of Cartesian products of cycles | B. Freyberg and M. KeranenReferences[1] S. Arumugam, D. Froncek and N. Kamatchi, Distance magic graphs - a survey, J. Indones.Math. Soc. (2011) Special edition, 11–26.[2] S. Cichacz, B. Freyberg and D. Froncek, Orientable Zn-distance magic graphs, Discuss.Math. (2017), accepted.[3] S. Cichacz and D. Froncek, Distance magic circulant graphs, Discrete Math. 339 (1) (2016),84–94.[4] B. Freyberg and M. Keranen, Orientable Zn-distance magic graphs via products, Preprint.311

Orientable ℤ \u3c inf\u3e n  -distance magic labeling of the Cartesian product of many cycles

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Orientable ℤ \u3c inf\u3e n -distance magic labeling of the Cartesian product of many cycles

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