A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of
a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian
group $\gr$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of
every vertex $x \in V$ is equal to the same element $\mu \in \gr$, called the
magic constant. In this paper we will show that if $G$ is a graph of order
$n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $\deg(v)\equiv c
\imod {2^{p+1}}$ for some constant $c$ for any $v\in V(G)$, then there exists
an $\gr$-distance magic labeling for any Abelian group $\gr$ for the graph
$G[C_4]$. Moreover we prove that if $\gr$ is an arbitrary Abelian group of
order $4n$ such that $\gr \cong \zet_2 \times\zet_2 \times \gA$ for some
Abelian group $\gA$ of order $n$, then exists a $\gr$-distance magic labeling
for any graph $G[C_4]$

D. Froncek

M. Miller

Sylwia Cichacz

English

arXiv

Crossref

Graphs and Combinatorics (2014) 30:565–571DOI 10.1007/s00373-013-1294-zORIGINAL PAPERNote on Group Distance Magic Graphs G[C4]Sylwia CichaczReceived: 10 May 2012 / Revised: 25 January 2013 / Published online: 24 February 2013© The Author(s) 2013. This article is published with open access at Springerlink.comAbstract A group distance magic labeling or a G-distance magic labeling of a graphG = (V, E) with |V | = n is a bijection f from V to an Abelian group G of ordern such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equal tothe same element μ ∈ G, called the magic constant. In this paper we will show thatif G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such thatdeg(v) ≡ c (mod 2p+1) for some constant c for any v ∈ V (G), then there exists aG-distance magic labeling for any Abelian group G of order 4n for the compositionG[C4]. Moreover we prove that if G is an arbitrary Abelian group of order 4n suchthat G ∼= Z2 × Z2 × A for some Abelian group A of order n, then there exists aG-distance magic labeling for any graph G[C4], where G is a graph of order n and nis an arbitrary natural number.Keywords Distance magic labeling · Magic constant · Sigma labeling · Graphlabeling · Abelian group · Composition of graphs · Lexicographic product of graphsMathematics Subject Classification (2000) 05C76 · 05C781 IntroductionAll graphs considered in this paper are simple finite graphs. Consider a simple graphG whose order we denote by |G| = n. Write V (G) for the vertex set and E(G) for theThe work was partially supported by National Science Centre grant nr 2011/01/D/ST/04104, as well as bythe Polish Ministry of Science and Higher Education.S. Cichacz (B)Faculty of Applied Mathematics, AGH University of Science and Technology,Al. Mickiewicza 30, 30-059 Kraków, Polande-mail: cichacz@agh.edu.pl123566 Graphs and Combinatorics (2014) 30:565–571edge set of the graph G. The neighborhood N (x) of a vertex x is the set of verticesadjacent to x , and the degree deg(x) of x is |N (x)|, the size of the neighborhood of x .A distance magic labeling (also called sigma labeling) of a graph G = (V, E) oforder n is a bijection l : V → {1, 2, . . . , n} with the property that there is a positiveinteger μ (called the magic constant) such that ∑y∈NG (x) l(y) = μ for every x ∈ V .If a graph G admits a distance magic labeling, then we say that G is a distance magicgraph ([14]). The sum∑y∈NG (x) l(y) is called the weight of the vertex x and denoted byw(x). The concept of distance magic labeling has been motivated by the constructionof magic squares.The following observations were independently proved:Observation 11 [10–12,14] Let G be an r-regular distance magic graph on n verti-ces. Then μ = r(n+1)2 .Observation 12 [10–12,14] No r-regular graph with r odd can be a distance magicgraph.The problem of distance magic labeling of r -regular graphs has been studiedrecently (see [2,3,6,11,13]). It is interesting that if you replace each vertex of anr -regular G graph by some specific p-regular graph (like C4 or K 2n ; cf. Defini-tion 13), then the obtained graph H is a distance magic graph. More formally, we havethe following definition:Definition 13 Let G and H be two graphs where {x1, x2, . . . , x p} is the vertex set ofG. Based upon the graph G, an isomorphic copy H j of H replaces every vertex x j ,for j = 1, 2, . . . , p, in such a way that a vertex in H j is adjacent to a vertex in Hi ifand only if x j xi was an edge in G. Let G[H ] denote the resulting graph. The graphG[H ] is called the composition of the graphs G and H [8].The graph G[H ] is also called the lexicographic product and denoted by G ◦ H(see [9]).Miller at al. [11] proved the following results.Theorem 14 [11] The cycle Cn of length n is a distance magic graph if and only ifn = 4.Theorem 15 [11] Let r ≥ 1, n ≥ 3, G be an r-regular graph and Cn be the cycle oflength n. The graph G[Cn] admits a distance magic labeling if and only if n = 4.Theorem 16 [11] Let G be an arbitrary regular graph. Then G[K n] is distance magicfor any even n.The following problem was posted in [2].Problem 17 [2] If G is a non-regular graph, determine if there is a distance magiclabeling of G[C4].It seems to be very hard to characterize such graphs. For example therewere considered all graphs Km,n[C4] for 1 ≤ m < n ≤ 2700 and onlyK9,21[C4], K20,32[C4], K428,548[C4] are distance magic graphs (see [1]).123Graphs and Combinatorics (2014) 30:565–571 567Froncek in [5] defined the notion of group distance magic graphs, i.e. the graphsallowing the bijective labeling of vertices with elements of an Abelian group resultingin constant sums of neighbor labels.Definition 18 A group distance magic labeling or a G-distance magic labeling of agraph G = (V, E) with |V | = n is a bijection f from V to an Abelian group G oforder n such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equalto the same element μ ∈ G, called the magic constant.Let G be a distance magic graph of order n with the magic constant μ′. If we replacethe label n in a distance magic labeling for the graph G by the label 0, then we obtain aZn-distance magic labeling for the graph G with the magic constant μ ≡ μ′ (mod n).Hence every distance magic graph with n vertices admits a Zn-distance magic label-ing. Although a Zn-distance magic graph on n vertices is not necessarily a distancemagic graph (see [5]), it was proved that Observation 12 also holds for a Zn-distancemagic labeling [4].Observation 19 [4] Let r be a positive odd integer. No r-regular graph on n verticescan be a Zn-distance magic graph.In this paper we will prove that if G is a graph of order n = 2p(2k + 1) for somenatural numbers p, k such that deg(v) ≡ c (mod 2p+1) for some constant c for anyv ∈ V (G), then there exists a G-distance magic labeling for any Abelian group G oforder 4n for the graph G[C4]. Moreover we show that if G is an Abelian group oforder 4n such that G ∼= Z2 × Z2 ×A for some Abelian group A of order n, then thereexists a G-distance magic labeling for any graph G[C4], where G is a graph of ordern and n is an arbitrary natural number.2 Main ResultsWe start with the following lemma.Lemma 21 Let G be a graph of order n and G be an arbitrary Abelian group of order4n such that G ∼= Z2p × A for p ≥ 2 and some Abelian group A of order n2p−2 . Ifdeg(v) ≡ c (mod 2p−1) for some constant c and any v ∈ V (G), then there exists aG-distance magic labeling for the graph G[C4].Proof Let G have the vertex set V (G) = {x0, x1, . . . , xn−1}, and let C4 = v0v1v2v3v0and H = G[C4].For j = 0, 1, 2, 3 let vij be the vertices of H that replace xi , i = 0, 1, . . . , n − 1,in G.Using the isomorphism φ : G → Z2p × A, we identify every g ∈ G with its imageφ(g) = (w, ai ), where w ∈ Z2p and ai ∈ A, i = 0, 1, . . . , n2p−2 − 1.123568 Graphs and Combinatorics (2014) 30:565–571Label the vertices of H in the following way:f (vij ) ={((2i + j) mod 2p−1, ai ·2−p+2)for j = 0, 1,(2p − 1, 0) − f (vij−2) for j = 2, 3for i = 0, 1, . . . , n − 1 and j = 0, 1, 2, 3.Notice that for every if (vi0) + f (vi2) = f (vi1) + f (vi3) = (2p − 1, 0).So the sum of the labels in the i th copy of C4 isf (vi0) + f (vi1) + f (vi2) + f (vi3) = (2p − 2, 0),which is independent on i . Since deg(v) ≡ c (mod 2p−1) for any v ∈ V (G), thereforethe weight of every x ∈ V (H) is w(x) = (−2c − 1, 0). unionsqTheorem 22 Let G be a graph of order n and G be an arbitrary Abelian group oforder 4n such that G ∼= Z2 × Z2 × A for some Abelian group A of order n. Thereexists a G-distance magic labeling for the graph G[C4].Proof Let V (G) = {x0, x1, . . . , xn−1} be the vertex set of G. Let C4 = v0v1v2v3v0and H = G[C4].For j = 0, 1, 2, 3 let vij be the vertices of H that replace xi , i = 0, 1, . . . , n − 1,in G.Using the isomorphism φ : G → Z2 × Z2 × A, we identify every g ∈ G with itsimage φ(g) = ( j1, j2, ai ), where j1, j2 ∈ Z2 and ai ∈ A, i = 0, 1, . . . , n − 1.Label the vertices of H in the following way:f (vij ) =⎧⎪⎪⎨⎪⎪⎩(0, 0, ai ) for j = 0,(1, 0, ai ) for j = 1,(1, 1,−ai ) for j = 2,(0, 1,−ai ) for j = 3for i = 0, 1, . . . , n − 1 and j = 0, 1, 2, 3.Notice that for every i = 0, . . . , n − 1f (vi0) + f (vi2) = f (vi1) + f (vi3) = (1, 1, 0).So the sum of the labels in the i th copy of C4 isf (vi0) + f (vi1) + f (vi2) + f (vi3) = (0, 0, 0),which is independent on i . Therefore, for every x ∈ V (H),w(x) = (1, 1, 0).unionsq123Graphs and Combinatorics (2014) 30:565–571 569Theorem 23 Let G be a graph of order n and G be an Abelian group of order 4n.If n = 2p(2k + 1) for some natural numbers p, k and deg(v) ≡ c (mod 2p+1) forsome constant c for any v ∈ V (G), then there exists a G-distance magic labeling forthe graph G[C4].Proof The fundamental theorem of finite Abelian groups states that the finite Abeliangroup G can be expressed as the direct sum of cyclic subgroups of prime-power order.This implies that G ∼= Z2α0 × Zpα11 × Zpα22 × . . . × Zpαmm for some α0 > 0, where4n = 2α0 ∏mi=1 pαii and pi for i = 1, . . . , m are not necessarily distinct primes.Suppose first that G ∼= Z2 × Z2 × A for some Abelian group A of order n, then weare done by Theorem 22. Observe now that the assumption deg(v) ≡ c (mod 2p+1)and the unique (additive) decomposition of any natural number c into powers of 2imply that there exist constants c1, c2, . . . , cp such that deg(v) ≡ ci (mod 2i ), fori = 1, 2, . . . , p, for any v ∈ V (G). Hence if G ∼= Z2α0 ×A for some 2 ≤ α0 ≤ p + 2and some Abelian group A of order 4n2α0 , then we obtain by Lemma 21 that there existsa G-distance magic labeling for the graph G[C4]. unionsqThe next observation follows easily from the above Theorem 23, however the belowObservation 25 shows an infinite family of Eulerian graphs with odd order such thatnone of these graphs is distance magic.Observation 24 Let G be a graph of odd order n and G be an Abelian group of order4n. If G is an Eulerian graph (i.e. all vertices of the graph G have even degrees), thenthere exists a G-distance magic labeling for the graph G[C4].Let us introduce the following definition. The Dutch windmill graph Dtm is thegraph obtained by taking t > 1 copies of the cycle Cm with a vertex c in common [7].Thus for t being even a graph Dt4 is an Eulerian graph of odd order 3t + 1. Let the i thcopy of a cycle C4 in Dt4 be cyi xi zi c, for i = 0, . . . , t − 1. Let now C4 = v0v1v2v3v0and H = Dt4[C4]. For j = 0, 1, 2, 3, let xij (yij , zij , c j resp.) be the vertices of H thatreplace xi (yi , zi , c, resp.) in Dt4, i = 0, 1, . . . , t − 1.Observation 25 There does not exist a distance magic graph Dt4[C4].Proof Suppose that l is a distance magic labeling of the graph Dt4[C4] and μ = w(x),for all vertices x ∈ V (Dt4[C4]). It is easy to observe that there exist natural numbersac, aix , aiy and aiz, 0 ≤ i ≤ t − 1, such that:– l(c0) + l(c2) = l(c1) + l(c3) = ac.– l(xi0) + l(xi2) = l(xi1) + l(xi3) = aix , for 0 ≤ i ≤ t − 1.– l(yi0) + l(yi2) = l(yi1) + l(yi3) = aiy , for 0 ≤ i ≤ t − 1.– l(zi0) + l(zi2) = l(zi1) + l(zi3) = aiz , for 0 ≤ i ≤ t − 1.Since aiz + 2aix + 2ac = w(zij ) = μ = w(yij ) = aiy + 2aix + 2ac, j = 0, 1, 2, 3,we obtain that aiz = aiy =: ai for 0 ≤ i ≤ t − 1. Moreover w(xij ) = aix + 4ai =w(yij ) = ai + 2aix + 2ac, j = 0, 1, 2, 3, hence aix = 3ai − 2ac for 0 ≤ i ≤ t − 1.Because of 7ai − 2ac = w(zij ) = μ = w(zi′j ) = 7ai′ − 2ac, j = 0, 1, 2, 3, we obtainai = ai ′ =: a, for all 0 ≤ i, i ′ ≤ t − 1.123570 Graphs and Combinatorics (2014) 30:565–571Since 7a − 2ac = μ = w(c j ) = ac + 4ta, for j = 0, 1, 2, 3, it has to be that3ac = (7 − 4t)a and therefore t = 1 (because ac, a > 0), a contradiction. unionsqThe following observation shows that in the general case the condition on thedegrees of the vertices v ∈ V (G) in Theorem 23 is not necessary for the existence ofa G-distance magic labeling of a graph G[C4]:Observation 26 Let K p,q be a complete bipartite graph with p even and q odd andlet G be an Abelian group of order 4(p +q). There exists a G-distance magic labelingfor the graph G[C4].Proof If G ∼= Z2 × Z2 × A for some Abelian group A of order n = p + q, thenthere exists a G-distance magic labeling for the graph K p,q [C4] by Theorem 22. Sup-pose now that G ∼= Z4 × A for some Abelian group A of order p + q. Let K p,qhave the partition vertex sets A = {x0, x1, . . . , x p−1}, B = {y0, y1, . . . , yq−1} andlet C4 = v0v1v2v3v0. For j = 0, 1, 2, 3, let xij be the vertices of K p,q [C4] thatreplace the vertex xi , 0 ≤ i ≤ p − 1, in K p,q , and ylj be the vertices that replaceyl , 0 ≤ l ≤ q − 1.Using the isomorphism φ : G → Z4 × A, we identify every g ∈ G with its imageφ(g) = ( j, ai ), where j ∈ Z4 and ai ∈ A, i = 0, 1, . . . , p + q − 1.Label the vertices of K p,q [C4] in the following way:f (xij ) ={(2 j, ai ) for j = 0, 1,(1, 0) − f (xij−2) for j = 2, 3for i = 0, 1, . . . , p − 1 and j = 0, 1, 2, 3.f (ylj ) ={(2 j, ap+l) for j = 0, 1,(3, 0) − f (ylj−2) for j = 2, 3for l = 0, 1, . . . , q − 1 and j = 0, 1, 2, 3.Notice thatf (xi0) + f (xi2) = f (xi1) + f (xi3) = (1, 0),f (yl0) + f (yl2) = f (yl1) + f (yl3) = (3, 0),for every i = 0, . . . , p − 1 and for every l = 0, . . . , q − 1.This implies:p−1∑i=0(∑3j=0 f (xij ))= p(2, 0) = (0, 0),q−1∑l=0(∑3j=0 f (ylj ))= q(2, 0) = (2, 0).Hence w(x) = (3, 0) for every x ∈ V (K p,q [C4]). unionsq123Graphs and Combinatorics (2014) 30:565–571 571Acknowledgments My sincere thanks are due to M. Anholcer and D. Froncek for help concerning thepreparation of this manuscript. I would like also to thank the anonymous reviewers for their helpful andconstructive comments that greatly contributed to improving the paper.Open Access This article is distributed under the terms of the Creative Commons Attribution Licensewhich permits any use, distribution, and reproduction in any medium, provided the original author(s) andthe source are credited.References1. Anholcer, M., Cichacz, S.: Note on distance magic graphs Km,n [C4], preprint2. Arumugam, S., Froncek, D., Kamatchi, N.: Distance Magic Graphs-A Survey, Journal of the IndonesianMathematical Society, Special Edition 1–9 (2011)3. Cichacz, S.: Distance magic (r, t)-hypercycles, Preprint MD 057 (2011)4. Cichacz, S., Froncek, D.: Distance magic circulant graphs, preprint5. Froncek, D.: Group distance magic labeling of Cartesian product of cycles, Aust. J. Combinat. 55,167–174 (2013)6. Froncek, D., Kovárˇ, P., Kovárˇová, T.: Constructing distance magic graphs from regular graphs, J. Com-bin. Math. Combin. Comput. 78, 349–354 (2011)7. Gallian, J.A.: A dynamic survey of graph labeling, Elec. J. Combin. DS6. http://www.combinatorics.org/Surveys/8. Harary, F.: Graph Theory, Reading, MA: Addison-Wesley, New York, p. 22, (1994)9. Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition, Wiley, New York (2000)10. Jinnah, M.I.: On -labelled graphs, In: Acharya, B.D., Hedge, S.M. (eds.) Technical Proceedings ofGroup Discussion on Graph Labeling Problems, pp.71-77 (1999)11. Miller, M., Rodger, C., Simanjuntak, R.: Distance magic labelings of graphs. Aust. J. Combin-at. 28, 305–315 (2003)12. Rao, S.B.: Sigma Graphs - A survey, In: Acharya, B.D., Arumugam, S., Rosa, A. (eds.) Labelings ofDiscrete Structures and Applications, Narosa Publishing House, New Delhi 135-140 (2008)13. Rao, S.B., Singh, T., Parameswaran, V.: Some sigma labelled graphs I, In: Arumugam, S.,Acharya,B.D., Rao, S.B. (eds.) Graphs, Combinatorics, Algorithms and Applications, Narosa Publishing House,New Delhi 125-133 (2004)14. Vilfred, V.: -labelled graph and Circulant Graphs, Ph.D. Thesis, University of Kerala, Trivandrum,India (1994)123

Graphs and Combinatorics

Note on Group Distance Magic Graphs G[C 4]

Springer - Publisher Connector

Graphs and Combinatorics (2014) 30:565–571
DOI 10.1007/s00373-013-1294-z
ORIGINAL PAPER
Note on Group Distance Magic Graphs G[C4]
Sylwia Cichacz
Received: 10 May 2012 / Revised: 25 January 2013 / Published online: 24 February 2013
© The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract A group distance magic labeling or a G-distance magic labeling of a graph
G = (V, E) with |V | = n is a bijection f from V to an Abelian group G of order
n such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equal to
the same element μ ∈ G, called the magic constant. In this paper we will show that
if G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such that
deg(v) ≡ c (mod 2p+1) for some constant c for any v ∈ V (G), then there exists a
G-distance magic labeling for any Abelian group G of order 4n for the composition
G[C4]. Moreover we prove that if G is an arbitrary Abelian group of order 4n such
that G ∼= Z2 × Z2 × A for some Abelian group A of order n, then there exists a
G-distance magic labeling for any graph G[C4], where G is a graph of order n and n
is an arbitrary natural number.
Keywords Distance magic labeling · Magic constant · Sigma labeling · Graph
labeling · Abelian group · Composition of graphs · Lexicographic product of graphs
Mathematics Subject Classification (2000) 05C76 · 05C78
1 Introduction
All graphs considered in this paper are simple finite graphs. Consider a simple graph
G whose order we denote by |G| = n. Write V (G) for the vertex set and E(G) for the
The work was partially supported by National Science Centre grant nr 2011/01/D/ST/04104, as well as by
the Polish Ministry of Science and Higher Education.
S. Cichacz (B)
Faculty of Applied Mathematics, AGH University of Science and Technology,
Al. Mickiewicza 30, 30-059 Kraków, Poland
e-mail: cichacz@agh.edu.pl
123
566 Graphs and Combinatorics (2014) 30:565–571
edge set of the graph G. The neighborhood N (x) of a vertex x is the set of vertices
adjacent to x , and the degree deg(x) of x is |N (x)|, the size of the neighborhood of x .
A distance magic labeling (also called sigma labeling) of a graph G = (V, E) of
order n is a bijection l : V → {1, 2, . . . , n} with the property that there is a positive
integer μ (called the magic constant) such that ∑y∈NG (x) l(y) = μ for every x ∈ V .
If a graph G admits a distance magic labeling, then we say that G is a distance magic
graph ([14]). The sum∑y∈NG (x) l(y) is called the weight of the vertex x and denoted by
w(x). The concept of distance magic labeling has been motivated by the construction
of magic squares.
The following observations were independently proved:
Observation 11 [10–12,14] Let G be an r-regular distance magic graph on n verti-
ces. Then μ = r(n+1)2 .
Observation 12 [10–12,14] No r-regular graph with r odd can be a distance magic
graph.
The problem of distance magic labeling of r -regular graphs has been studied
recently (see [2,3,6,11,13]). It is interesting that if you replace each vertex of an
r -regular G graph by some specific p-regular graph (like C4 or K 2n ; cf. Defini-
tion 13), then the obtained graph H is a distance magic graph. More formally, we have
the following definition:
Definition 13 Let G and H be two graphs where {x1, x2, . . . , x p} is the vertex set of
G. Based upon the graph G, an isomorphic copy H j of H replaces every vertex x j ,
for j = 1, 2, . . . , p, in such a way that a vertex in H j is adjacent to a vertex in Hi if
and only if x j xi was an edge in G. Let G[H ] denote the resulting graph. The graph
G[H ] is called the composition of the graphs G and H [8].
The graph G[H ] is also called the lexicographic product and denoted by G ◦ H
(see [9]).
Miller at al. [11] proved the following results.
Theorem 14 [11] The cycle Cn of length n is a distance magic graph if and only if
n = 4.
Theorem 15 [11] Let r ≥ 1, n ≥ 3, G be an r-regular graph and Cn be the cycle of
length n. The graph G[Cn] admits a distance magic labeling if and only if n = 4.
Theorem 16 [11] Let G be an arbitrary regular graph. Then G[K n] is distance magic
for any even n.
The following problem was posted in [2].
Problem 17 [2] If G is a non-regular graph, determine if there is a distance magic
labeling of G[C4].
It seems to be very hard to characterize such graphs. For example there
were considered all graphs Km,n[C4] for 1 ≤ m < n ≤ 2700 and only
K9,21[C4], K20,32[C4], K428,548[C4] are distance magic graphs (see [1]).
123
Graphs and Combinatorics (2014) 30:565–571 567
Froncek in [5] defined the notion of group distance magic graphs, i.e. the graphs
allowing the bijective labeling of vertices with elements of an Abelian group resulting
in constant sums of neighbor labels.
Definition 18 A group distance magic labeling or a G-distance magic labeling of a
graph G = (V, E) with |V | = n is a bijection f from V to an Abelian group G of
order n such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equal
to the same element μ ∈ G, called the magic constant.
Let G be a distance magic graph of order n with the magic constant μ′. If we replace
the label n in a distance magic labeling for the graph G by the label 0, then we obtain a
Zn-distance magic labeling for the graph G with the magic constant μ ≡ μ′ (mod n).
Hence every distance magic graph with n vertices admits a Zn-distance magic label-
ing. Although a Zn-distance magic graph on n vertices is not necessarily a distance
magic graph (see [5]), it was proved that Observation 12 also holds for a Zn-distance
magic labeling [4].
Observation 19 [4] Let r be a positive odd integer. No r-regular graph on n vertices
can be a Zn-distance magic graph.
In this paper we will prove that if G is a graph of order n = 2p(2k + 1) for some
natural numbers p, k such that deg(v) ≡ c (mod 2p+1) for some constant c for any
v ∈ V (G), then there exists a G-distance magic labeling for any Abelian group G of
order 4n for the graph G[C4]. Moreover we show that if G is an Abelian group of
order 4n such that G ∼= Z2 × Z2 ×A for some Abelian group A of order n, then there
exists a G-distance magic labeling for any graph G[C4], where G is a graph of order
n and n is an arbitrary natural number.
2 Main Results
We start with the following lemma.
Lemma 21 Let G be a graph of order n and G be an arbitrary Abelian group of order
4n such that G ∼= Z2p × A for p ≥ 2 and some Abelian group A of order n2p−2 . If
deg(v) ≡ c (mod 2p−1) for some constant c and any v ∈ V (G), then there exists a
G-distance magic labeling for the graph G[C4].
Proof Let G have the vertex set V (G) = {x0, x1, . . . , xn−1}, and let C4 = v0v1v2v3v0
and H = G[C4].
For j = 0, 1, 2, 3 let vij be the vertices of H that replace xi , i = 0, 1, . . . , n − 1,
in G.
Using the isomorphism φ : G → Z2p × A, we identify every g ∈ G with its image
φ(g) = (w, ai ), where w ∈ Z2p and ai ∈ A, i = 0, 1, . . . , n2p−2 − 1.
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568 Graphs and Combinatorics (2014) 30:565–571
Label the vertices of H in the following way:
f (vij ) =
{(
(2i + j) mod 2p−1, a
i ·2−p+2
)
for j = 0, 1,
(2p − 1, 0) − f (vij−2) for j = 2, 3
for i = 0, 1, . . . , n − 1 and j = 0, 1, 2, 3.
Notice that for every i
f (vi0) + f (vi2) = f (vi1) + f (vi3) = (2p − 1, 0).
So the sum of the labels in the i th copy of C4 is
f (vi0) + f (vi1) + f (vi2) + f (vi3) = (2p − 2, 0),
which is independent on i . Since deg(v) ≡ c (mod 2p−1) for any v ∈ V (G), therefore
the weight of every x ∈ V (H) is w(x) = (−2c − 1, 0). unionsq
Theorem 22 Let G be a graph of order n and G be an arbitrary Abelian group of
order 4n such that G ∼= Z2 × Z2 × A for some Abelian group A of order n. There
exists a G-distance magic labeling for the graph G[C4].
Proof Let V (G) = {x0, x1, . . . , xn−1} be the vertex set of G. Let C4 = v0v1v2v3v0
and H = G[C4].
For j = 0, 1, 2, 3 let vij be the vertices of H that replace xi , i = 0, 1, . . . , n − 1,
in G.
Using the isomorphism φ : G → Z2 × Z2 × A, we identify every g ∈ G with its
image φ(g) = ( j1, j2, ai ), where j1, j2 ∈ Z2 and ai ∈ A, i = 0, 1, . . . , n − 1.
Label the vertices of H in the following way:
f (vij ) =
⎧
⎪⎪⎨
⎪⎪⎩
(0, 0, ai ) for j = 0,
(1, 0, ai ) for j = 1,
(1, 1,−ai ) for j = 2,
(0, 1,−ai ) for j = 3
for i = 0, 1, . . . , n − 1 and j = 0, 1, 2, 3.
Notice that for every i = 0, . . . , n − 1
f (vi0) + f (vi2) = f (vi1) + f (vi3) = (1, 1, 0).
So the sum of the labels in the i th copy of C4 is
f (vi0) + f (vi1) + f (vi2) + f (vi3) = (0, 0, 0),
which is independent on i . Therefore, for every x ∈ V (H),
w(x) = (1, 1, 0).
unionsq
123
Graphs and Combinatorics (2014) 30:565–571 569
Theorem 23 Let G be a graph of order n and G be an Abelian group of order 4n.
If n = 2p(2k + 1) for some natural numbers p, k and deg(v) ≡ c (mod 2p+1) for
some constant c for any v ∈ V (G), then there exists a G-distance magic labeling for
the graph G[C4].
Proof The fundamental theorem of finite Abelian groups states that the finite Abelian
group G can be expressed as the direct sum of cyclic subgroups of prime-power order.
This implies that G ∼= Z2α0 × Zpα11 × Zpα22 × . . . × Zpαmm for some α0 > 0, where
4n = 2α0 ∏mi=1 pαii and pi for i = 1, . . . , m are not necessarily distinct primes.
Suppose first that G ∼= Z2 × Z2 × A for some Abelian group A of order n, then we
are done by Theorem 22. Observe now that the assumption deg(v) ≡ c (mod 2p+1)
and the unique (additive) decomposition of any natural number c into powers of 2
imply that there exist constants c1, c2, . . . , cp such that deg(v) ≡ ci (mod 2i ), for
i = 1, 2, . . . , p, for any v ∈ V (G). Hence if G ∼= Z2α0 ×A for some 2 ≤ α0 ≤ p + 2
and some Abelian group A of order 4n2α0 , then we obtain by Lemma 21 that there exists
a G-distance magic labeling for the graph G[C4]. unionsq
The next observation follows easily from the above Theorem 23, however the below
Observation 25 shows an infinite family of Eulerian graphs with odd order such that
none of these graphs is distance magic.
Observation 24 Let G be a graph of odd order n and G be an Abelian group of order
4n. If G is an Eulerian graph (i.e. all vertices of the graph G have even degrees), then
there exists a G-distance magic labeling for the graph G[C4].
Let us introduce the following definition. The Dutch windmill graph Dtm is the
graph obtained by taking t > 1 copies of the cycle Cm with a vertex c in common [7].
Thus for t being even a graph Dt4 is an Eulerian graph of odd order 3t + 1. Let the i th
copy of a cycle C4 in Dt4 be cyi xi zi c, for i = 0, . . . , t − 1. Let now C4 = v0v1v2v3v0
and H = Dt4[C4]. For j = 0, 1, 2, 3, let xij (yij , zij , c j resp.) be the vertices of H that
replace xi (yi , zi , c, resp.) in Dt4, i = 0, 1, . . . , t − 1.
Observation 25 There does not exist a distance magic graph Dt4[C4].
Proof Suppose that l is a distance magic labeling of the graph Dt4[C4] and μ = w(x),
for all vertices x ∈ V (Dt4[C4]). It is easy to observe that there exist natural numbers
ac, a
i
x , a
i
y and aiz, 0 ≤ i ≤ t − 1, such that:
– l(c0) + l(c2) = l(c1) + l(c3) = ac.
– l(xi0) + l(xi2) = l(xi1) + l(xi3) = aix , for 0 ≤ i ≤ t − 1.
– l(yi0) + l(yi2) = l(yi1) + l(yi3) = aiy , for 0 ≤ i ≤ t − 1.
– l(zi0) + l(zi2) = l(zi1) + l(zi3) = aiz , for 0 ≤ i ≤ t − 1.
Since aiz + 2aix + 2ac = w(zij ) = μ = w(yij ) = aiy + 2aix + 2ac, j = 0, 1, 2, 3,
we obtain that aiz = aiy =: ai for 0 ≤ i ≤ t − 1. Moreover w(xij ) = aix + 4ai =
w(yij ) = ai + 2aix + 2ac, j = 0, 1, 2, 3, hence aix = 3ai − 2ac for 0 ≤ i ≤ t − 1.
Because of 7ai − 2ac = w(zij ) = μ = w(zi
′
j ) = 7ai
′ − 2ac, j = 0, 1, 2, 3, we obtain
ai = ai ′ =: a, for all 0 ≤ i, i ′ ≤ t − 1.
123
570 Graphs and Combinatorics (2014) 30:565–571
Since 7a − 2ac = μ = w(c j ) = ac + 4ta, for j = 0, 1, 2, 3, it has to be that
3ac = (7 − 4t)a and therefore t = 1 (because ac, a > 0), a contradiction. unionsq
The following observation shows that in the general case the condition on the
degrees of the vertices v ∈ V (G) in Theorem 23 is not necessary for the existence of
a G-distance magic labeling of a graph G[C4]:
Observation 26 Let K p,q be a complete bipartite graph with p even and q odd and
let G be an Abelian group of order 4(p +q). There exists a G-distance magic labeling
for the graph G[C4].
Proof If G ∼= Z2 × Z2 × A for some Abelian group A of order n = p + q, then
there exists a G-distance magic labeling for the graph K p,q [C4] by Theorem 22. Sup-
pose now that G ∼= Z4 × A for some Abelian group A of order p + q. Let K p,q
have the partition vertex sets A = {x0, x1, . . . , x p−1}, B = {y0, y1, . . . , yq−1} and
let C4 = v0v1v2v3v0. For j = 0, 1, 2, 3, let xij be the vertices of K p,q [C4] that
replace the vertex xi , 0 ≤ i ≤ p − 1, in K p,q , and ylj be the vertices that replace
yl , 0 ≤ l ≤ q − 1.
Using the isomorphism φ : G → Z4 × A, we identify every g ∈ G with its image
φ(g) = ( j, ai ), where j ∈ Z4 and ai ∈ A, i = 0, 1, . . . , p + q − 1.
Label the vertices of K p,q [C4] in the following way:
f (xij ) =
{
(2 j, ai ) for j = 0, 1,
(1, 0) − f (xij−2) for j = 2, 3
for i = 0, 1, . . . , p − 1 and j = 0, 1, 2, 3.
f (ylj ) =
{
(2 j, ap+l) for j = 0, 1,
(3, 0) − f (ylj−2) for j = 2, 3
for l = 0, 1, . . . , q − 1 and j = 0, 1, 2, 3.
Notice that
f (xi0) + f (xi2) = f (xi1) + f (xi3) = (1, 0),
f (yl0) + f (yl2) = f (yl1) + f (yl3) = (3, 0),
for every i = 0, . . . , p − 1 and for every l = 0, . . . , q − 1.
This implies:
p−1∑
i=0
(∑3
j=0 f (x
i
j )
)
= p(2, 0) = (0, 0),
q−1∑
l=0
(∑3
j=0 f (y
l
j )
)
= q(2, 0) = (2, 0).
Hence w(x) = (3, 0) for every x ∈ V (K p,q [C4]). unionsq
123
Graphs and Combinatorics (2014) 30:565–571 571
Acknowledgments My sincere thanks are due to M. Anholcer and D. Froncek for help concerning the
preparation of this manuscript. I would like also to thank the anonymous reviewers for their helpful and
constructive comments that greatly contributed to improving the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution License
which permits any use, distribution, and reproduction in any medium, provided the original author(s) and
the source are credited.
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