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ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 13 (2017) 417-425
On zero sum-partition of Abelian groups into three sets and group distance magic labeling
Sylwia Cichacz
Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
Received 2 March 2016, accepted 1 May 2017, published online 10 May 2017
Abstract
We say that a finite Abelian group r has the constant-sum-partition property into t sets (CSP(t)-property) if for every partition n = ri + r2 + ... + rt of n, with r® > 2 for 2 < i < t, there is a partition of r into pairwise disjoint subsets Ai,A2,..., At, such that |Aj| = r and for some v G r, J2aeA- a = v for 1 < i < t. For v = g0 (where g0 is the identity element of r) we say that r has zero-sum-partition property into t sets (ZSP (t) -property).
A r-distance magic labeling of a graph G = (V, E) with | V| = n is a bijection I from V to an Abelian group r of order n such that the weight w(x) = J2yeN(x) ^(v) of every vertex x G V is equal to the same element p G r, called the magic constant. A graph G is called a group distance magic graph if there exists a r-distance magic labeling for every Abelian group r of order |V(G)|.
In this paper we study the CSP(3)-property of r, and apply the results to the study of group distance magic complete tripartite graphs.
Keywords: Abelian group, constant sum partition, group distance magic labeling. Math. Subj. Class.: 05C25, 05C78
1 Introduction
All graphs considered in this paper are simple finite graphs. Consider a simple graph G whose order we denote by n = |G|. We denote by V(G) the vertex set and E(G) the edge set of a graph G. The open neighborhood N(x) of a vertex x is the set of vertices adjacent to x, and the degree d(x) of x is |N(x) |, the size of the neighborhood of x.
E-mail address: cichacz@agh.edu.pl (Sylwia Cichacz)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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Let the identity element of r be denoted by g0. Recall that any group element i G r of order 2 (i.e., i = g0 such that 2i = g0) is called an involution.
In [8] Kaplan, Lev and Roditty introduced a notion of zero-sum partitions of subsets in Abelian groups. Let r be an Abelian group and let A be a finite subset of r - {g0}, with | A| = n — 1. We shall say that A has the zero-sum-partition property (ZSP-property) if every partition n — 1 = ri + r2 + ... + rt of n — 1, with r > 2 for 1 < i < t and for any possible positive integer t, there is a partition of A into pairwise disjoint subsets A1, A2,..., At, such that |Aj| = r and J2aaA- a = g0 for 1 < i < t. In the case that r is finite, we shall say that r has the ZSP-property if A = r — {g0} has the ZSP-property.
They proved the following theorem for cyclic groups of odd order.
Theorem 1.1 ([8]). The group Zn has the ZSP-property if and only if n is odd.
Moreover, Kaplan, Lev and Roditty showed that if r is a finite Abelian group of even order n such that the number of involutions in r is different from 3, then r does not have the ZSP-property [8]. Their results along with results proved by Zeng [10] give necessary and sufficient conditions for the ZSP-property for a finite Abelian group.
Theorem 1.2 ([8, 10]). Let r be a finite Abelian group. Then r has the ZSP-property if and only if either r is of odd order or r contains exactly three involutions.
They apply those results to the study of anti-magic trees [8, 10].
We generalize the notion of ZSP-property. We say that a finite Abelian group r has the constant-sum-partition property into t sets (CSP(t)-property) if for every partition n = r1 + r2 + ... + rt of n, with r > 2 for 2 < i < t, there is a partition of r into pairwise disjoint subsets A1, A2,..., At, such that |Aj| = r and for some v G r, J2aeAi a = v for 1 < i < t. For v = g0 we say that r has zero-sum-partition property into t sets (ZSP(t)-property).
In this paper we investigate also distance magic labelings, which belong to a large family of magic type labelings.
A distance magic labeling (also called sigma labeling) of a graph G = (V, E) of order n is a bijection I: V ^ {1, 2,..., n} with the property that there is a positive integer k (called the magic constant) such that
w(x) = ^^ ¿(y) = k for every x G V(G),
yEN (x)
where w(x) is the weight of vertex x. If a graph G admits a distance magic labeling, then we say that G is a distance magic graph.
The concept of distance magic labeling has been motivated by the construction of magic rectangles, since we can construct a distance magic complete r-partite graph with each part size equal to n by labeling the vertices of each part by the columns of the magic rectangle. Although there does not exist a 2 x 2 magic rectangle, observe that the partite sets of K2 2 can be labeled {1,4} and {2,3}, respectively, to obtain a distance magic labeling. The following result was proved in [9].
Observation 1.3 ([9]). There is no distance magic r-regular graph with r odd.
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Froncek in [7] defined the notion of group distance magic graphs, i.e., the graphs allowing a bijective labeling of vertices with elements of an Abelian group resulting in constant sums of neighbor labels.
A T-distance magic labeling of a graph G = (V, E) with | V| = n is a bijection I from V to an Abelian group r of order n such that the weight w(x) = J2yeN(x) ^(y) of every vertex x e V is equal to the same element m e r, called the magic constant. A graph G is called a group distance magic graph if there exists a T-distance magic labeling for every Abelian group T of order |V(G)|.
The connection between distance magic graphs and T-distance magic graphs is as follows. Let G be a distance magic graph of order n with the magic constant M. 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 m = M (mod n). Hence every distance magic graph with n vertices admits a Zn-distance magic labeling. However a Zn-distance magic graph on n vertices is not necessarily a distance magic graph. Moreover, there are some graphs that are not distance magic while at the same time they are group distance magic (see [4]).
A general theorem for T-distance magic labeling similar to Observation 1.3 was proved recently.
Theorem 1.4 ([5]). Let G be an r-regular graph on n vertices, where r is odd. There does not exist an Abelian group T of order n with exactly one involution i such that G is T-distance magic.
Notice that the constant sum partitions of a group T lead to complete multipartite T-distance magic labeled graphs. For instance, the partition {0}, {1,2,4}, {3,5,6} of the group Z7 with constant sum 0 leads to a Z7 -distance magic labeling of the complete tripartite graph K1i3i3. More general, let G be a complete t-partite graph of order n with the partition sets V1, V2,..., Vt. Note that G is T-distance magic if and only if Ei=i,i=j Exev ^(x) = M for j G {1, 2,...,t} which implies to	^(x) = v for
j G {1,2,...,t} and some v e T. Therefore we can see that G is T-distance magic if and only if r has the CSP(t)-property. The following theorems were proven in [3].
Theorem 1.5 ([3]). Let G = Kni,n2,...,nt be a complete t-partite graph and n = n1 + n2 + ... + nt. If n = 2 (mod 4) and t is even, then there does not exist an Abelian group r of order n such that G is a T-distance magic graph.
Theorem 1.6 ([3]). The complete bipartite graph Kni,n2 is a group distance magic graph if and only if n1 + n2 ^ 2 (mod 4).
Therefore it follows that an Abelian group T of order n has the CSP(2)-property if and only if n ^ 2 (mod 4).
In this paper we study the CSP(3)-property of T, and apply the results to an investigation of the necessary and sufficient conditions for complete tripartite graphs to be group distance magic. This work will also be potentially useful for group theorists working on Abelian groups.
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2	Preliminaries
Assume r is an Abelian group of order n with the operation denoted by +. For convenience we will write ka to denote a + a + ... + a (where the element a appears k times), —a to denote the inverse of a and we will use a - b instead of a + (—b). Recall that a non-trivial finite group has elements of order 2 if and only if the order of the group is even. The fundamental theorem of finite Abelian groups states that a finite Abelian group r of order n can be expressed as the direct product of cyclic subgroups of prime-power order. This implies that
r = Zpa x Zp^2 x ... x Zp^k where n = p^1 • p22 • ... • p£k
and pi for i e {1, 2,..., k} are not necessarily distinct primes. This product is unique up to the order of the direct product. When t is the number of these cyclic components whose order is a multiple of 2, then r has 2l — 1 involutions. In particular, if n = 2 (mod 4), then r = Z2 x A for some Abelian group A of odd order n/2. Moreover every cyclic group of even order has exactly one involution. The sum of all the group elements is equal to the sum of the involutions and the neutral element.
The following lemma was proved in [6] (see [6], Lemma 8).
Lemma 2.1 ([6]). Let r be an Abelian group.
1.	If r has exactly one involution i, thenY g£r 9 = u
2.	If r has no involutions, or more than one involution, then Yger 9 = 9o.
Anholcer and Cichacz proved the following (see [1], Lemma 2.4).
Lemma 2.2 ([1]). Let r be an Abelian group with involutions set I* = {ti, t2,..., i2k k > 1 and let I = I * U {90}. Given positive integers n\, n2 such that ni + n2 = 2k. There exists a partition A = {Ai, A2} of I such that
1.	ni = |Ai|, n2 = |A2|,
2.	EaeA, a = 90 for i = 1, a if and only if none of ni, n2 is 2.
3	Constant sum partition of Abelian groups
Note that if r has odd order, then it has the ZSP-property by Theorem 1.2, thus one can check that it has the ZSP(3)-property. We now generalize Lemma 2.2.
Lemma 3.1. Let r be an Abelian group with involutions set I * = {ti, t2,..., t2fc_i}, k > 2 andlet I = I* U{g0 }. Given positive integers ni, n2, n3 suchthat ni+n2+n3 = 2k. There exists a partition A = {Ai, A2, A3} of I such that
1.	ni = |Ai|, n2 = |A2|, n3 = |A3|,
2.	EaeA, a = 90 for i e {1, 2, 3}, if and only if ni,n2,n3 e {2, 2k — 2}.
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Proof. For n = n = 1 we have that J2o£a a = 2»ea a. For n = 2, it is easy to see
EaeAi a = g0.
Let i0 = g0. Recall that since I = {i0, f,..., t2fc-i} is a subgroup of r, we have I = (Z2)k. One can check that the lemma is true for k e {3,4}. The sufficiency will be proved then by induction on k. Namely, suppose the assertion is true for some m = k > 4. We want to prove it is true for m = k + 1. Let (n1, n2, n3) be a triple such that
n1,n2,n3 £ {2, 2k+1 - 2} and n1 + n2 + n3 = 2fc+1. For i e {l, 2, 3} let n = 4qj + r, where r e {1,3,4, 5,6} and 1 appears at most once as a value of some r. Observe
that r1 + r2 + r3 < 18, but because n1 + n2 + n3 = 0 (mod 4) and n1 + n2 + n3 = 4(q1 + q2 + q3) + r1 + r2 + r3, we must have r1 + r2 + r3 = 0 (mod 4), which implies that r1 + r2 + r3 < 16. Thus 4(q1 + q2 + 93) > 2k.
Now we select t1,t2,t3 such that tj < qj and 4(t1 +t2 +t3) = 2k .Denote nj = n-4tj and nj' = 4tj. Obviously, n1 + n2 + n3 = n'/ + n2' + n3' = 2k and nj e {0, 2, 2k - 2} for any i e {1, 2, 3}.
If also n'' = 0, then both triples (n1, n2, n3) and (n", n2', n3') satisfy the inductive hypothesis and there exist partitions of (Z2)k into sets S', S2, S3 and S'', S2', S3' of respective orders n1, n2, n3 and n1', n2', n3'. If we now replace each element (x1, X2,..., x^) of (Z2)k in any Sj by the (x1, x2,..., xk, 0) of (Z2)k+1, it should be clear that the sum of elements in each Sj is the identity of (Z2)k+1.
Similarly, we replace each element (y, y2,..., ) of (Z2)k in any Sj' by the element (y, y2,..., yk, 1) of (Z2)k+1. Now because the order of each Sj' is even, the ones in last entries add up to zero and the sum of elements in each S'' is again the identity of (Z2)k+1. Now set Sj = Sj U S'' to obtain the desired partition of (Z2)k+1.
The case when n'' = 0 and nj', n" = 0 can be treated using Lemma 2.2, and the case when n'' = nj' = 0 and n" = 2k is obvious.	□
Theorem 3.2. Let r be an Abelian group of even order n. r has the CSP(3) -property if and only if r (Z2)4 for some positive integer t. Moreover, r = (Z2)4 has the ZSP(3)-property if and only if r has more than one involution.
Proof. For a given partition n = n +n2+n3 we will construct a partition r = A1 UA2 UA3 such that Aj = {a0, «1,..., a^._1} for i G {1, 2, 3}. Let r = {g0, g1,..., gn-1}. Recall that by g0 we denote the identity element of r.
Assume first that r = (Z2)4 for t > 1 has the CSP(3)-property. Let Ai, A2, A3 be the desired partition of r for n2 =2. Hence J2aeA a = 1 = g0 for i G {1,2,3}. Therefore Eger g = J23=1 SaeA « = 3i = 1 = g0, a contradiction with Lemma 2.1.
Suppose now that r has the ZSP(3)-property and there is the only one involution 1 G r. Let A1, A2, A3 be the desired partition of r, therefore J2« = g0 for i G {1, 2,3}. Hence, g0 = J23=1 J2aeAi « = Sser g, on the other hand by Lemma 2.1 we have Soer g = 1, a contradiction.
We will prove sufficiency now. Let us consider two cases on the number of involutions in r.
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Case 1. There is exactly one involution i in r.
Notice that in that case |r| > 6. By fundamental theorem of finite Abelian groups
r = Z2"1 x X ... x Zp^k where n = 2ai • pf2 • ... • pf, «1 > 1
and pi > 3 for i e {2,3,..., k} are not necessarily distinct primes. Since |r| > 6 we have r = Z2m xA for m > 3 and some Abelian group A of order n/2m. Let g1 = i and
gi+1 = —gi for i e {2,4, 6,..., n - 2}. Using the isomorphism ^: r ^ Z2m xA, we can identify every g e r with its image ^(g) = (j, ai), where j e Z2m and ai e A for i e {0,1,..., 2m — 1} and a0 is the identity element in A. Observe that g1 = i = (m, a0). Because m > 2 we can set g2 = (1,a0), g3 = (2m — 1,a0), g4 = (m — 1, a0), g5 = (m + 1,ao).
Without loss of generality we can assume that n1 is even and n2 > n3. Let aj = g2,
a1 = g4 and a1 = gi+4 for i e {2, 3,..., n1 — 1}.
Case 1.1. n2, n3 are both odd.
Let: a2 = go, af = g3, a2 = g5 and g1 = a„i+1+i for i e {3,4,..., n — 1}. a0 = g1 and a3 = g„i+„2+i for i e {1,2,... — 1}.
Case 1.2. n2, n3 are both even.
Let: a2 = g3, a2 = g5 and a1 = g„1+2+i for i e {2, 3,..., n — 1}. a0 = go, a3 = g1 and a3 = gm +„2+i for i e {2, 3,... — 1}.
Note that in both Cases 1.1 and 1.2 we obtain that J2aeA a = (m, a0) = i for
i e {1, 2, 3}.
Case 2. There is more that one involution i in r.
By fundamental theorem of finite Abelian groups r has 2* — 1 involutions i1, i2,..., i2t_1 for t > 1. Let gi = ii for i e {1, 2,..., 2* — 1}, and gi+1 = —gi for i e {2*, 2* + 2, 2* + 4,..., n — 2}. By the above arguments on necessity we obtain that r = (Z2)*, therefore 2* < n/2. One can check, that we can choose integers t1, t2 and t3 such that:
t1 + t2 + t3 =2*,
with
ni — ti = 0 (mod 2), ti > 0, ti e {2, 2* — 2} for i e {1, 2, 3}.
By Lemmas 2.2 and 3.1 it follows that there exists a partition B = {B1, B2, B3} of
I = {go,g1,. ..,g2t_1} such that t1 = |B11, t2 = |B21, t3 = |Bs|, and if Bi = 0, then b = go for i e {1,2, 3}. Let Bi = {b0, bj,..., biii_1} for i e {1,2, 3}. Let us set
now:
a1 = b1 for i e {1, 2,... ,t1 — 1} and a1 = gi+i2+i3 for i e {t1,t1 + 1..., n1 — 1}, a2 = b2 for i e {1, 2,... ,t2 — 1} and a2 = gi+i3+ni for i e {t2,t2 + 1..., n2 — 1}, a3 = b3 for i e {1, 2,... ,t3 — 1} and a3 = gi+„1+„2 for i e {t3 ,t3 + 1... ,n3 — 1}. In this case Eo£a a = g0 for i e {1,2,3}.	□
4 Group distance magic graphs
Observe that for G being an odd regular graph of order n, by hand shaking lemma n is even. Thus, the below theorem is a generalization of Theorem 1.4.
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Theorem 4.1. Let G have order n = 2 (mod 4) with all vertices having odd degree. There does not exist an Abelian group r of order n such that G is a r-distance magic graph.
Proof. Assumption n = 2 (mod 4) implies that r = Z2 xA for some Abelian group A of odd order n/2 and there exists exactly one involution i e r. Let gn/2 = i, gn/2+i = —a for i e {1, 2, ...,n/2 — 1}. Let V (G) = {x0,x^... ,xn-1}.
Suppose that I is a r-distance labeling for G and ^ is the magic constant. Without loss of generality we can assume that i(xi) — a for i e {0,1,..., n — 1}. Recall that ng = 0 for any g e r and deg(xn/2)gn/2 = gn/2 = i since deg(xn/2) is odd. Notice that deg(xi) — deg(xn-i) = 2di for some integer di for i e {1,2,..., n/2 — 1}, because all vertices have odd degree. Let now
n-1
w(G) =	w(y) = J^deg(xi)gi =
xeV(G) yeN(x)	i=0
n/2-1	n/2-1
E deg(xi)gi + deg(x„/2)g„/2 + E deg(x„-i)g„-i = i=1 i=1
n/2-1	n/2-1
^2 deg(xi)gi — ^2 deg(xn-i)gi + gn/2 = i=1 i=1
n/2-1	n/2-1
E (deg( xi) — deg(xn-i))gi + gn/2 = 2 E digi + gn/2 i=1 i=1
On the other hand, w(G) = J2xeV(G) w(x) = n • ^ = g0. Therefore we obtain that 2u = gn/2 for some element u e r. Since n/2 is odd and r = Z2 x A, such an element u does not exist, a contradiction.	□
From the above Theorem 4.1 we obtain the following.
Theorem 4.2. If G have order n = 2 (mod 4) with all vertices having odd degree, then G is not distance magic.
Proof. The graph G is not Zn-distance magic by Theorem 4.1, therefore it is not distance magic.	□
We prove now the following useful lemma.
Lemma 4.3. Let G = Kni,n2,...,nt be a complete t-partite graph andn = n1 + n2 +... + nt. If n1 < n2 < ... < nt and n2 = 1, then there does not exist an Abelian group r of order n such that G is a T-distance magic graph.
Proof. Let G have the partition vertex sets Vi such that | V | = ni for i e {1, 2,..., t}. Let x e V1 and y e V2. Suppose that the graph G is r-distance magic for some Abelian group r of order n and that I is a r-distance magic labeling of G, then w(x) = J2ger g — ^(x) = w(y) = J2ger g — %). Thus ¿(y) = ^(x), a contradiction.	□
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Theorem 4.4. Let G = Kni,n2,n3 be a complete tripartite graph such that 1 < n < n2 < n3 and n = ni + n2 + n3. The graph G is a group distance magic graph if and only if n2 > 1 and n1 + n2 + n3 = 2p for any positive integer p.
Proof. Let G have the partition vertex sets V such that |Vj| = n» for i G {1, 2,3}. We can assume that n2 > 1 by Lemma 4.3.
Suppose now that r = (Z2 )p for some integer p. Let n1 = 2 and £ be a r-distance magic labeling of G. Thus ExeVl £(x) = i = go. Since G is r-distance magic we obtain
that Ex€v, £(x) = i for i G {1,2,3}. Therefore Eser g = E3=1 Ex€v, £(x) = 3i = i = g0, a contradiction with Lemma 2.1.
If r ^ (Z2)p and n > 2 for i G {2, 3}, then the group r can be partitioned into pair-wise disjoint sets A1, A2, A3 such that for every i G {1,2, 3}, |A» | = n» with E o£a a = v for some element v G r by Theorem 1.2 or 3.2. Label the vertices from a vertex set V using elements from the set A» for i G {1,2,3}.	□
Theorem 4.5. Let G = Kni,n2,n3 be a complete tripartite graph such that 1 < n1 < n2 < n3 and n1 + n2 + n3 = 2p, then
1.	G is r-distance magic for any Abelian group r = (Z2)p of order n if and only if
n2 > 1 ,
2.	G is (Z2)p -distance magic if and only if n1 = 2 and n2 > 2.
Proof. Let G have the partition vertex sets V such that |Vi| = n» for i G {1,2, 3}.
We can assume that n2 > 1 by Lemma 4.3. If (n1 = 2 or n2 > 2) and r = (Z2)p then r does not have a partition A = {A1, A2, A3} such that EaeA- a = v for i G {1,2,3} by Theorem 3.2. Thus one can check that then there does not exist a r-distance labeling of G.
If r ^ (Z2)p and ni > 2 for i G {2, 3}, or r ^ (Z2)p for some integer p and n1 = 2, n2 > 2, then the group r can be partitioned into pairwise disjoint sets A1, A2, A3 such that for every i G {1,2,3}, |Aj| = n4 with Ea€A. a = v for some element v G r by Theorem 3.2, or Lemma 3.1, resp. Label the vertices from a vertex set V using elements from the set A» for i G {1, 2,3}.	□
At the end of this section we put some observations that are implications of Theorem 1.2 for complete t-partite graphs. But first we need the following theorem proved in [2] (see Theorem 2.2, [2]).
Theorem 4.6 ([2]). Let G be a graph for which there exists a distance magic labeling £: V(G) ^ {1,2,..., |V(G)|} such that for every w G V(G) the following holds: if u G N(w) with £(u) = i, then there exists v G N(w), v = u, with £(v) = |V(G)| + 1 — i. The graph G is a group distance magic graph.
Observation 4.7. Let G = Kni,n2,...,nt be a complete t-partite graph such that 1 < n1 < n2 < ... < nt and n = n1 + n2 + ... + nt. Let r be an Abelian group of order n with exactly three involutions. The graph G is r-distance magic graph if and only if n2 > 1.
Proof. Let G have the partition vertex sets V = {x1, x2,..., x^.} for i G {1, 2,..., t}. By Lemma 4.3 we can assume that n2 > 1.
Suppose first that n1 = n2 = ... = nt = 2. Note that a labeling £: V (G) ^{1,2,..., 2t} defined as £(x1) = i, £(x|) = 2t + 1 — i for i G {1,2,..., t} is distance magic, hence G is
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a group distance magic graph by Theorem 4.6. This implies that there exists a T-distance magic labeling of G.
We can assume now that nt > 3. If n > 1, then nt > 4 or nt-1 = nt = 3. Therefore there exists a zero-sum partition A1, A2,..., At of the set r - {go } such that | At | = nt -1 and |Ai | = n for every 1 < i < t - 1 by Theorem 1.2. Set At = At U {go} and Aj = Aj for every 1 < i < t — 1. If ni = 1 then there exists a zero-sum partition A2, A3,..., At of the set r - {g0} such that |Aj| = n for every 2 < i < t by Theorem 1.2. In this case put A1 = {g0} and A» = Aj for every 2 < i < t. Label now the vertices from a vertex set Vj using elements from the set Aj for i € {1, 2,..., t}.	□
Observation 4.8. Let G = Kni i„2 be a complete t-partite graph such that 1 < n1 < n2 < ... < nt and n = n1 + n2 + ... + nt is odd. The graph G is a group distance magic graph if and only if n2 > 1.
Proof. Let G have the partition vertex sets Vj such that |Vj| = n for i € {1, 2,..., t}. We can assume that n2 > 1 by Lemma 4.3. If n1 > 1, then nt > 3. Therefore there exists a zero-sum partition A1, A2,..., At of the set r - {g0} such that | At | = nt - 1 and |Aj| = n, for every 1 < i < t - 1 by Theorem 1.2. Set At = At U {g0} and A, = Aj for every 1 < i < t - 1. If n1 = 1 then there exists a zero-sum partition A2, A3,..., At of the set r - {g0} such that |Aj| = n for every 2 < i < t by Theorem 1.2. In this case put A1 = {g0} and Aj = Aj for every 2 < i < t. Label now the vertices from a vertex set Vj using elements from the set Aj for i € {1, 2,..., t}.	□
Acknowledgments. I thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.
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