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\def\ck{\mathop{\overline{K}_3}\nolimits}
\def\cck{\mathop{\overline{K}_2}\nolimits}
%D \def\mod{\mathop{\rm mod}\nolimits}
\def\dim{\mathop{\rm dim}\nolimits}
\newcommand{\ent}{\mathbb N}
\newcommand{\Ge}{\mathfrak{G}}
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\def\gcd{\mathop{\rm gcd}\nolimits}
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\title{\bf Zero sum partition of Abelian groups into sets \\of the same
order and its applications}%D

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\author{Sylwia Cichacz\\
\small Faculty of Applied Mathematics\\[-0.8ex]
\small AGH University of Science and Technology\\[-0.8ex]
\small Al. Mickiewicza 30, 30-059 Krak\'ow, Poland\\
\small\tt cichacz@agh.edu.pl}
%\small Mathematics Subject Classifications: 05C25, 05C78}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list off,
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\date{\dateline{Apr 19, 2017}{Jan 15, 2018}{Jan 25, 2018}\\
\small Mathematics Subject Classifications: 05C25, 05C78}

\begin{document}

\maketitle



\begin{abstract}
  We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-\textit{ZSP-property}) if  $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots, A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, where $g_0$ is the identity element of $\Gamma$. 


In this paper we study the $m$-ZSP property of $\Gamma$. We show that $\Gamma$ has the $m$-ZSP property if and only if $m\geq 3$ and $|\Gamma|$ is odd or   $\Gamma$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.



  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Abelian group, zero sum partition,  group distance magic labeling, Kotzig arrays
\end{abstract}

\section{Introduction}
Assume $\Gamma$ is an Abelian group of order $n$ with the operation denoted by $+$.  For convenience
we will write $ka$ to denote $a + a + \cdots + a$ (where the element $a$ appears $k$ times), $-a$ to denote the inverse of $a$ and
we will use $a - b$ instead of $a+(-b)$.  Moreover, the notation $\sum_{a\in S}{a}$ will be used as a short form for $a_1+a_2+a_3+\cdots$, where $a_1, a_2, a_3, \dots$ are all elements of the set $S$. The identity element of $\Gamma$ will be denoted by $g_0$. Recall that any group element $\iota\in\Gamma$ of order 2 (i.e., $\iota\neq g_0$ and $2\iota=g_0$) is called an \emph{involution}.

In \cite{KLR} Kaplan, Lev and  Roditty introduced the notion of zero-sum partitions of subsets in Abelian groups. Let $\Gamma$ be an Abelian group and let $A$ be a finite subset of $\Gamma - \{g_0\}$, with $|A| = n-1$. We shall say that $A$ has the
\emph{zero-sum-partition property} (\textit{ZSP-property}) if for every partition $n-1 = r_1 + r_2 + \cdots + r_t$ of $n-1$, with $r_i \geq 2$ for $1 \leq i \leq t$ and for any possible positive integer $t$, there is a partition of $A$ into pairwise disjoint subsets $A_1, A_2,\ldots, A_t$, such that $|A_i| = r_i$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$. When $\Gamma$ is finite, we shall say that $\Gamma$ has the ZSP-property if $A = \Gamma - \{g_0\}$ has the ZSP-property.

The following theorem for cyclic groups  was proved in~\cite{KLR}.
\begin{theorem}[\cite{KLR}] The group $\zet_n$ has the ZSP-property if and only if $n$ is odd.
\end{theorem}
 Moreover, Kaplan, Lev and  Roditty showed that if  $\Gamma$ is a finite Abelian group of an even order $n$ such that the number of involutions in $\Gamma$ is different from 3, then $\Gamma$ does not have the ZSP-property \cite{KLR}. Their results along with results proved by  Zeng \cite{Zeng}
 give necessary and sufficient conditions for the ZSP-property for  a finite Abelian group.
\begin{theorem}[\cite{KLR, Zeng}] Let $\Gamma$ be a finite Abelian group. Then $\Gamma$ has the ZSP-property
if and only if either $\Gamma$ is of an odd order or $\Gamma$ contains exactly three involutions. \label{ZSP}
\end{theorem}
They used those results to  study anti-magic trees \cite{KLR, Zeng}. In \cite{AnhCic} the ZSP-property of cyclic groups of an odd order was used for determining the group irregularity strength of disconnected graphs.\\
 

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-\textit{ZSP-property}) if  $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots, A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$. \\


A similar problem, where the orders of subsets are not important, is called the \textit{modular sumset partition problem}. This problem was studied in \cite{ref_LlaMor2}.  A non-increasing sequence $\left\langle m_1,\ldots, m_k\right\rangle$ of positive integers is said to be $n$-\textit{realizable} if the set $\{1,2,\ldots,n\}$ can be partitioned into $k$ mutually disjoint subsets $X_1, X_2 \ldots,X_k$ such that $\sum_{x\in X_i}x = m_i$ for each $1 \leq i \leq k$.  Llad\'o and Moragas considered the modular version of the problem and, by using the polynomial method introduced by Alon \cite{ref_Alon}, they proved that all sequences in $\zet_p$ of length $k\leq(p-1)/2$ are realizable for any prime $p\geq 3$ \cite{ref_LlaMor2}. The study of $n$-realizable sequences was motivated by the ascending subgraph
decomposition problem posed by Alavi, Boals, Chartrand, Erd\H{o}s and Oellerman \cite{ref_AlaBoaChaErd}.





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 $G$. The \emph{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 order of the neighborhood of $x$. In this paper we  also investigate  distance magic labelings, which belong to a large family of magic-type labelings. Generally speaking, a magic-type labeling of a graph $G=(V,E)$ is a mapping from $V, E,$ or $V\cup E$ to a set of labels which most often is a set of integers or group elements. Then the weight of a graph element is typically the sum of labels of the neighboring elements of one or both types. When the weight of each element is required to be equal, then we speak about magic-type labeling; when the weights are all different (or even form an arithmetic progression), then we speak about an antimagic-type labeling. Probably the best known problem in this area is the {\em antimagic conjecture} by Hartsfield and Ringel~\cite{HR}, which claims that the edges of every graph except $K_2$ can be labeled by integers $1,2,\dots,|E|$ so that the weight of each vertex is different.

A \emph{distance magic labeling} (also called \emph{sigma
labeling}) of a graph $G=(V,E)$ of order $n$ is a bijection $\ell
\colon V \rightarrow \{1, 2,\ldots, n\}$ with the property that
there is a positive integer $k$  such
that
%
$$w(x)=\sum_{y\in N(x)}\ell(y)=k~ {\text{for every}}~ x \in V(G).$$
%
If a graph $G$ admits a distance magic labeling, then we say that $G$ is a \emph{distance magic graph}. O'Neal and Slater obtained a formula for the vertex magic constant in
terms of a fractional domination parameter of the graph, which implies the
uniqueness of the magic constant \cite{ONeSla}. The following result was proved in \cite{MRS}.
\begin{observation}[\cite{MRS}] There is no distance magic $r$-regular graph with $r$ odd.\label{nieparzyste}\end{observation}




Froncek in \cite{Fro} 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 $\Gamma$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$
such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to the same element $\mu\in \Gamma$, called the \emph{magic
constant}. A graph $G$ is called a \emph{group distance magic graph} if there exists a $\Gamma$-distance magic labeling for every Abelian
group $\Gamma$ of order $|V(G)|$.\\


The connection between distance magic graphs and $\Gamma$-distance magic graphs is the following. Let $G$ be a distance magic
graph of order $n$ with the magic constant $\mu'$. If we replace the label $n$ in a distance magic labeling for the graph $G$
by the label $0$, then we obtain a $\zet_n$-distance magic labeling for the graph $G$ with the magic constant $\mu \equiv\mu'\pmod n$.
Hence every distance magic graph with $n$ vertices admits a $\zet_n$-distance magic labeling. However, a $\zet_n$-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 \cite{Cic2}).

A general theorem for $\Gamma$-distance magic labeling similar to Observation~\ref{nieparzyste} was proved recently.
\begin{theorem}[\cite{CicFro}]\label{gr:odd} Let $G$ be an $r$-regular graph on $n$
vertices, where $r$ is odd.
There does not exist an Abelian group $\Gamma$ of order $n$ with exactly one
involution $\iota$ such that $G$ is $\Gamma$-distance magic.
\end{theorem}

The following theorem was proved in \cite{Cic3}.
\begin{theorem}[\cite{Cic3}]\label{2mod4_1} Let $G$ be a graph of order $n \equiv 2 \pmod 4$
with all vertices of odd degree.
Then there is no Abelian group $\Gamma$ of order $n$ such that $G$ is a $\Gamma$-distance magic graph.
\end{theorem}

Notice that the constant sum partitions of a group $\Gamma$ lead to complete multipartite $\Gamma$-distance magic labeled graphs. For
instance, the partition $\{0\}$, $\{1, 2, 4\}$, $\{3, 5, 6\}$ of the group $\zet_7$ with constant sum $0$ leads to a $\zet_7$-distance magic labeling
of the complete tripartite graph $K_{1,3,3}$ (see \cite{Cic2,Cic3}).


The paper is organized as follows.  In Section 2 we give some preliminaries on Abelian groups. In Section~\ref{sZSP}  we show that $\Gamma$ has the $m$-ZSP property if and only if $|\Gamma|$ is odd or $\Gamma$ has more than one involution.
So far there was  only one known example of a $\Gamma$-distance magic labeling of an odd regular graph for a group $\Gamma$ which has more than one involution, namely $K_{3,3,3,3}$ with $\Gamma\cong\zet_2\times\zet_2\times\zet_3$ (see~\cite{CicFro}). In Section~\ref{GDM} we use the $m$-ZSP property of Abelian groups to show an infinite family of odd regular graphs possessing $\Gamma$-distance magic labeling for groups $\Gamma$ with more than one involution.  Finally, in Section~ \ref{Kotzig} we introduce a generalization of Kotzig arrays and  give necessary and sufficient conditions for their existence.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55555

\section{Preliminaries}

 A non-trivial
finite group has elements of order $2$ (involutions) if and only if the order of the group is even. The fundamental theorem of finite Abelian groups states that a finite Abelian
group $\Gamma$ of order $n$ can be expressed as the direct product of cyclic subgroups of prime-power order. This implies that
$$\Gamma\cong\zet_{p_1^{\alpha_1}}\times\zet_{p_2^{\alpha_2}}\times\cdots\times\zet_{p_k^{\alpha_k}}\;\;\; \mathrm{where}\;\;\; n = p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$$
and $p_i$ for $i \in \{1, 2,\ldots,k\}$ are not necessarily distinct primes. This product is unique up to the order of the direct product. When $p$ is the number of these cyclic components whose order is a multiple of $2$, then $\Gamma$ has $2^p-1$ involutions. In particular, if $n \equiv 2 \pmod 4$, then $\Gamma\cong \zet_2\times \Lambda$ for some
Abelian group $\Lambda$ of odd order $n/2$.  Moreover every cyclic group of an even order has exactly one involution. Since the properties and results in this paper are invariant under the isomorphism between groups, we only need to consider one group in each isomorphism class. 

The sum of all elements of a group $\Gamma$ is equal to the sum of its involutions and the identity element. The following lemma was proved in~\cite{CN} (see \cite{CN}, Lemma 8).


\begin{lemma}[\cite{CN}]\label{involutions} Let $\Gamma$  be an Abelian group.
\begin{itemize}
 \item[-] If $\Gamma$ has exactly one involution $\iota$, then $\sum_{g\in \Gamma}g= \iota$.
\item[-] If $\Gamma$ has no involutions, or more than one involution, then $\sum_{g\in \Gamma}g=g_0$.
\end{itemize}
\end{lemma}

Zeng proves a lemma which plays an important role in the proof of the
main result  (see \cite{Zeng}, Lemma 2.1).


\begin{lemma}[\cite{Zeng}]\label{bijection}
Let $\Gamma$ be a finite Abelian group of an odd order or $\Gamma$ contains exactly three involutions. Let Bij$(\Gamma)$ denote the set of all bijections from $\Gamma$ to itself.
Then there exist $\phi,\varphi\in$Bij$(\Gamma)$ (not necessarily distinct) such that $g+\phi(g)+\varphi(g)=g_0$  for every $g\in\Gamma$. %In particular, we mayassumethat #(0)='(0)=0.
\end{lemma}


%%%%%%%%%%%%%%%%
\section{Zero sum partition}\label{sZSP}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let the identity element of  a group $\Lambda$ be denoted by $a_0$. By $\langle g \rangle$ we denote the  subgroup generated by $g$ in the group $\Gamma$. A quotient group $\Gamma$ modulo $H$ for a subgroup $H$ of $\Gamma$ will be denoted by $\Gamma/H$.\\

For convenience, let $\gr$ denote the set consisting of all Abelian groups which are
of odd order or contain more than one involution.

We will start with some lemmas.


\begin{lemma}
\label{involutions3} Let $\Gamma$  be an Abelian group  with involution set $I^* =\{\iota_1,\iota_2,\ldots,\iota_{2^p-1}\}$, $p >1$ and let $I = I^*\cup\{g_0\}$.
 Then there exists a partition $A=\{A_1, A_2,\ldots, A_{2^{p-2}}\}$ of $I$ such that $|A_i|=4$,
   $\sum_{a\in A_i}a=g_0$ for $i\in\{1,2,\ldots,2^{p-2}\}$.\end{lemma}

\begin{proof}
Let $\iota_0=g_0$. Recall that  $I=\{\iota_0,\iota_1,\ldots,\iota_{2^p-1}\}$ is a subgroup of $\Gamma$. Moreover, notice that $\Lambda=\{\iota_0,\iota_1,\iota_2,\iota_1+\iota_2\}$ is a subgroup of $I$. Therefore there exists a coset decomposition of $I$ into $\iota_{j_1}+\Lambda,\iota_{j_2}+\Lambda,\ldots,\iota_{j_{2^{p-2}}}+\Lambda$ for  $\iota_{j_k}\in I$, $k=1,2,\ldots,2^{p-2}$. Set $A_k=\iota_{j_k}+\Lambda$ for $k=1,2,\ldots,2^{p-2}$. Obviously $\sum_{\iota\in A_k}\iota=\iota_0$.
\end{proof}


Using the same method as in the proof of Lemma \ref{bijection}, we can obtain the following lemma.
\begin{lemma}\label{bijections}
Let $\Gamma$ be a finite Abelian group of odd order or $\Gamma$ contains more than one involution. Let Bij$(\Gamma)$ denote the set of all bijections from $\Gamma$ to itself.
Then there exist $\phi,\varphi\in$Bij$(\Gamma)$ (not necessarily distinct) such that $g+\phi(g)+\varphi(g)=g_0$  for every $g\in\Gamma$. %In particular, we mayassumethat #(0)='(0)=0.
\end{lemma}
\begin{proof}
First we prove the following claim. Suppose that $\Gamma_1,\Gamma_2\in\gr$ and the assertion of 
the lemma is true for $\Gamma_1,\Gamma_2$. Then the assertion holds for $\Gamma=\Gamma_1\times\Gamma_2$. Assume the bijections for $\Gamma_1$ are $\phi_1$ and $\varphi_1$, whereas those for $\Gamma_2$ are $\phi_2$ and $\varphi_2$. Then
$$\phi=(\phi_1,\phi_2): \Gamma_1\times \Gamma_2\rightarrow\Gamma_1\times \Gamma_2,\;\; (a_1,a_2)\mapsto (\phi_1(a_1),\phi_2(a_2))$$
and
$$\varphi=(\varphi_1,\varphi_2): \Gamma_1\times \Gamma_2\rightarrow\Gamma_1\times \Gamma_2,\;\; (a_1,a_2)\mapsto (\varphi_1(a_1),\varphi_2(a_2)).$$
By the assertion above and Lemma~\ref{bijection}, and noting that this lemma is invariant under  isomorphism, it suffices to prove it for $\Gamma=\zet_{2^\alpha}\times\zet_{2^\beta}\times\zet_{2^\kappa}$  with $\alpha,\beta,\kappa\geq1$.

The proof is by induction on $|\Gamma|$. We deal with three base cases.  For $\zet_{2}\times\zet_{2}\times\zet_{2}$   the table below  gives the desired bijections.\\
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
% \diagbox{$h$}{$j$}&0& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline
$g$          & (0,0,0) & (0,0,1) & (0,1,0) & (0,1,1) & (1,0,0) & (1,0,1) & (1,1,0) & (1,1,1)\\ \hline
$\phi(g)$    & (0,0,1) & (1,1,1) & (0,1,0) & (1,0,0) & (0,0,0) & (1,1,0) & (0,1,1) & (1,0,1)\\ \hline
$\varphi(g)$ & (0,0,1) & (1,1,0) & (0,0,0) & (1,1,1) & (1,0,0) & (0,1,1) & (1,0,1) & (0,1,0)\\ \hline
\end{tabular}\newline


Whereas  for  $\zet_{2}\times\zet_{2}\times\zet_{4}$  the table:\\
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
% \diagbox{$h$}{$j$}&0& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline
$g$          & (0,0,0) & (0,1,0) & (1,0,0) & (1,1,0) & (0,0,1) & (0,1,1) & (1,0,1) & (1,1,1)\\ \hline
$\phi(g)$    & (0,1,0) & (1,1,2) & (1,0,1) & (0,0,3) & (1,0,3) & (0,0,1) & (0,1,3) & (1,1,1)\\ \hline
$\varphi(g)$ & (0,1,0) & (1,0,2) & (0,0,3) & (1,1,1) & (1,0,0) & (0,1,2) & (1,1,0) & (0,0,2)\\\hline\hline
$g$          & (0,0,2) & (0,1,2) & (1,0,2) & (1,1,2) & (0,0,3) & (0,1,3) & (1,0,3) & (1,1,3)\\ \hline
$\phi(g)$    & (0,0,2) & (1,0,0) & (1,1,3) & (0,1,1) & (0,0,0) & (1,0,2) & (1,1,0) & (0,1,2)\\ \hline
$\varphi(g)$ & (0,0,0) & (1,1,2) & (0,1,3) & (1,0,1) & (0,0,1) & (1,1,3) & (0,1,1) & (1,0,3)\\ \hline
\end{tabular}
\newline\newline


Suppose that $\zet_{2}\times\zet_{2}\times\zet_{8}$. Then for $g=(i,j,0)$ and $g=(i,j,4)$ we use the table for $\zet_{2}\times\zet_{2}\times\zet_{2}$ where in the last coordinate we put 4 instead of $1$.\\


If $g=(i,j,a)$ for $a \not \in\{0,4\}$, then set a triple $$(a,b,c)\in \{(2,3,3),(7,2,7),(5,6,5),(6,1,1),(1,5,2),(3,7,6)\},$$ and for $g=(i,j,a)$ we have the following table.\\
\begin{tabular}{|c|c|c|c|c|}
\hline$g$    & (0,0,a) & (0,1,a) & (1,0,a) & (1,1,a) \\ \hline
$\phi(g)$    & (0,0,b) & (1,0,b) & (1,1,b) & (0,1,b) \\ \hline
$\varphi(g)$ & (0,0,c) & (1,1,c) & (0,1,c) & (1,0,c) \\ \hline

\end{tabular}\newline

Now we proceed to the induction part. 
Suppose first that $\Gamma=\zet_{2^\alpha}\times\zet_{2^\beta}\times\zet_{2^\kappa}$  with $\beta,\kappa\geq2$. Then there exists a subgroup $\Gamma_0=\langle 1 \rangle\times \langle 2\rangle\times \langle 2 \rangle\in\gr$ of $\Gamma$ such that $\Gamma/\Gamma_0\cong \zet_2\times\zet_2$.
By the induction hypothesis, there are $\phi_0$, $\varphi_0\in$Bij$(\Gamma_0)$ such that $a+\phi_0(a)+\varphi_0(a)=g_0$ for every $a\in \Gamma_0$. Choose a set of coset representatives for $\Gamma/\Gamma_0$ to be $\{0,c,d,-c-d\}$. Note that
$$(c+\Gamma_0)\bigcup(d+\Gamma_0)\bigcup(-c-d+\Gamma_0)=\bigcup_{b\in\Gamma_0}\{c+b,d+\phi_0(b),-c-d+\varphi_0(b)\},$$
and every subset $\{c+b,d+\phi_0(b),-c-d+\varphi_0(b)\}$ is zero-sum. 
Therefore $\Gamma=\Gamma_0\cup(\cup_{i=1}^{|\Gamma_0|}\{a_{i,1},a_{i,2},a_{i,3}\})$ where  $a_{i,1}+a_{i,2}+a_{i,3}=g_0$ for $1\leq i\leq |\Gamma_0|$. Thus define $\phi$ and
$\varphi$ as follows.
$$\phi(a)=\left\{\begin{array}{ccl}
\phi_0(a) & \mathrm{if} & a\in\Gamma_0, \\
                     %\phi_1(a) & \mathrm{if} & a\in\Gamma_1, \\
											a_{i,2} & \mathrm{if}& a=a_{i,1}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq|\Gamma_0|, \\
											a_{i,3} & \mathrm{if}& a=a_{i,2} \;\;\mathrm{for}\;\; \rm{some}\;\;1\leq i\leq|\Gamma_0|, \\
											a_{i,1} & \mathrm{if} &a=a_{i,3}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq|\Gamma_0|, \\
 \end{array}\right.$$
and
$$\varphi(a)=\left\{\begin{array}{ccl}
                    \varphi_0(a) & \mathrm{if} & a\in\Gamma_0, \\
											a_{i,3} & \mathrm{if} &a=a_{i,1}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq|\Gamma_0|, \\
											a_{i,1} & \mathrm{if} &a=a_{i,2} \;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq|\Gamma_0|, \\
											a_{i,2} & \mathrm{if}& a=a_{i,3} \;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq|\Gamma_0|. \\
 \end{array}\right.$$


Suppose now that $\Gamma=\zet_{2}\times\zet_{2}\times\zet_{2^\kappa}$. The cases when $\kappa \leq 3$ have been shown in the base case. Thus we may assume that  $\kappa\geq 4$ and there exists a subgroup $\Gamma_0=\langle 1 \rangle\times \langle 1 \rangle\times \langle 8 \rangle\in\gr$ such that $\Gamma/\Gamma_0\cong \zet_8$. By the induction hypothesis, there are $\phi_0$, $\varphi_0\in$Bij$(\Gamma_0)$ such that $a+\phi_0(a)+\varphi_0(a)=g_0$ for every $a\in \Gamma_0$. Since
$$\zet_8=\{0,4\}\bigcup\{1,2,-3\}\bigcup\{-1,-2,3\},$$
we can choose a set of coset representatives for $\Gamma/\Gamma_0$ to be $\{0,e,c,d,-c-d,-c,-d,c+ d\}$ with $2e\in \Gamma_0$ (for example $e=(0,0,4)$). Therefore $\Gamma_1=\Gamma_0\cup (e+\Gamma_0)$ (for example if $e=(0,0,4)$  then $\Gamma_1=\langle 1 \rangle\times \langle 1 \rangle\times \langle 4 \rangle$) is a group and moreover  $\Gamma_1\in\gr$.
%
%
Thus there exist $\phi_1,\varphi_1\in$Bij$(\Gamma_1)$ such that $a+\phi_1(a)+\varphi_1(a)=g_0$ for every $a\in \Gamma_1$. Similarly as above, we have
$$(c+\Gamma_0)\bigcup(d+\Gamma_0)\bigcup(-c-d+\Gamma_0)=\bigcup_{b\in\Gamma_0}\{c+b,d+\phi_0(b),-c-d+\varphi_0(b)\},$$
and
$$(-c+\Gamma_0)\bigcup(-d+\Gamma_0)\bigcup(c+d+\Gamma_0)=\bigcup_{b\in\Gamma_0}\{-c+b,-d+\phi_0(b),c+d+\varphi_0(b)\}.$$
Hence $\Gamma=\Gamma_1\cup(\cup_{i=1}^{2|\Gamma_0|}\{a_{i,1},a_{i,2},a_{i,3}\})$ where  $a_{i,1}+a_{i,2}+a_{i,3}=g_0$ for $1\leq i\leq 2|\Gamma_0|$. Define $\phi$ and
$\varphi$ as follows.
$$\phi(a)=\left\{\begin{array}{ccl}
                     \phi_1(a) & \mathrm{if} & a\in\Gamma_1, \\
											a_{i,2} & \mathrm{if}& a=a_{i,1}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq2|\Gamma_0|, \\
											a_{i,3} & \mathrm{if}& a=a_{i,2} \;\;\mathrm{for}\;\; \rm{some}\;\;1\leq i\leq2|\Gamma_0|, \\
											a_{i,1} & \mathrm{if} &a=a_{i,3}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq2|\Gamma_0|, \\
 \end{array}\right.$$
and
$$\varphi(a)=\left\{\begin{array}{ccl}
                    \varphi_1(a) & \mathrm{if} & a\in\Gamma_1, \\
											a_{i,3} & \mathrm{if} &a=a_{i,1}\;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq2|\Gamma_0|, \\
											a_{i,1} & \mathrm{if} &a=a_{i,2} \;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq2|\Gamma_0|, \\
											a_{i,2} & \mathrm{if}& a=a_{i,3} \;\;\mathrm{for}\;\; \rm{some}\;\; 1\leq i\leq2|\Gamma_0|. \\
 \end{array}\right.$$
This completes the proof.\end{proof}

\begin{lemma}\label{mZSPeven} An Abelian group $\Gamma$  of order $n$ has the $(2m)$-ZSP-property  if and only if $(2m)|n$, $m\geq2$ and $\Gamma\in\gr$.
\end{lemma}
\begin{proof} We will show the necessity first. Suppose that $\Gamma$  has the $(2m)$-ZSP-property and $A_1, A_2,\ldots, A_t$ is the desired partition. Therefore the order of $\Gamma$ is even and divisible by $2m$. Hence $\Gamma$ has at least one involution. Obviously $m>1$. Assume now that there is exactly one involution $\iota\in \Gamma$. On one hand $\sum_{g\in \Gamma}g=\sum_{i=1}^t\sum_{a\in A_i}a=g_0$, on the other $\sum_{g\in \Gamma}g=\iota$ by Lemma~\ref{involutions}, a contradiction. 

 

Let $\Gamma$  be an Abelian group  with involution set $I^* =\{\iota_1,\iota_2,\ldots,\iota_{2^p-1}\}$, $p >1$  and let $I = I^*\cup\{g_0\}$. Then there exists a partition $\{I_1, I_2,\ldots, I_{2^{p-2}}\}$ of $I$ such that $|I_i|=4$,
   $\sum_{a\in I_i}a=g_0$ for $i\in\{1,2,\ldots,2^{p-2}\}$ by Lemma  \ref{involutions3}.
	Note that all elements in the set $\Gamma- I$ can be split into disjoint pairs $B_i=\{g_i,-g_i\}$ for $i=1,2,\ldots,(|\Gamma|-2^p)/2$. 


It is easy to see now that for $2m\equiv 0 \pmod 4$ there exists a partition $A=\{A_1, A_2,\ldots,$ $A_{|\Gamma|/2m}\}$ of $\Gamma$ such that $|A_i|=2m$ and 
   $\sum_{a\in A_i}a=g_0$ for $i\in\{1,2,\ldots,|\Gamma|/2m\}$. 

Recall that $\Gamma$ can be expressed as a direct product of cyclic subgroups of prime-power order.  When $p$ is the number of these cyclic components
whose order is a power of $2$, then $\Gamma$ has $2^p-1$ involutions.	
Note that for $2m\equiv 2 \pmod 4$, because $m\geq3$ is odd we obtain $\Gamma\cong\zet_{2^{\alpha_1}}\times\zet_{2^{\alpha_2}}\times\ldots\times\zet_{2^{\alpha_p}}\times\Lambda$ for $\alpha_1,\alpha_2,\ldots,\alpha_p\geq1$ and some Abelian group $\Lambda$ of an odd order divisible by $m$. Therefore $(|\Gamma|-2^p)=|\Gamma- I|\geq (m-1)|I|=(m-1)2^p$. Hence $(|\Gamma|-2^p)/2\geq2^p>2^{p-2}=|I|/4$ and one can check that there exists a partition $A=\{A_1, A_2,\ldots, A_{|\Gamma|/2m}\}$ of $\Gamma$ such that $|A_i|=2m$,
   $\sum_{a\in A_i}a=g_0$ for $i\in\{1,2,\ldots,|\Gamma|/2m\}$.\end{proof}

\begin{lemma}\label{mZSPodd} An Abelian group $\Gamma$  of order  $n$ has the $(2m+1)$-ZSP-property  if and only if  $m\geq1$, $(2m+1)|n$ and $\Gamma\in\gr$.
\end{lemma}
\begin{proof} We will show the necessity first. The assumptions  $m\geq1$ and $(2m+1)|n$ are obvious. Suppose that $\Gamma$  has exactly one involution $\iota$ and the $(2m+1)$-ZSP-property. Let $A_1, A_2,\ldots, A_t$ be the desired partition. On one hand $\sum_{g\in \Gamma}g=\sum_{i=1}^t\sum_{a\in A_i}a=g_0$, on the other $\sum_{g\in \Gamma}g=\iota$ by Lemma~\ref{involutions}, a contradiction. 

Assume that $|\Gamma|$ is odd. Let $t=n/(2m+1)$. There exists a partition $A_1',A_2',\ldots,A_t'$ of the set $\Gamma- \{g_0\}$ such that $|A_1'|=2m$ and for every $2 \leq i \leq t$, $|A_i'| =2m+1$ with $\sum_{a\in A_i'} a =g_0$ by Theorem~\ref{ZSP}.  We obtain the desired partition $A_1,A_2,\ldots,A_t$ by setting $A_1=A_1'\cup \{g_0\}$ and $A_i=A_i'$ for every $2 \leq i \leq t$.


Suppose now that $\Gamma$ has  $2^{p}-1$ involutions for some positive integer $p>1$. Therefore $\Gamma\cong\Gamma_0\times\Lambda$ for  $\Gamma_0\cong\zet_{2^{\alpha_1}}\times\zet_{2^{\alpha_2}}\times\cdots\times\zet_{2^{\alpha_p}}$ for $\alpha_1,\alpha_2,\ldots,\alpha_p\geq1$ and some Abelian group $\Lambda$  of odd order divisible by $(2m+1)$. Let $t=|\Lambda|/(2m+1)$. Obviously $\Gamma_0,\Lambda\in\gr$.  Hence $\Lambda$ has the $(2m+1)$-ZSP-property and there  exists a partition $A_1,A_2,\ldots,$ $A_{t}$ of the group $\Lambda$ such that for every $1 \leq i \leq t$, $|A_i| = 2m+1$ with $\sum_{a\in A_i} a =a_0$. Denote the $h$-th element in the set $A_i$ by $a_{i,h}$.

Define a function $\gamma\colon\Gamma_0\rightarrow\Gamma_0$ such that $\gamma(w)=(2m-1)w$ for all $w\in\Gamma_0$. Note that  $\gamma\in$Bij$(\Gamma_0)$ (and moreover it is an automorphism), because $\gcd(2^{\alpha},2m-1)=1$ for any positive integer $\alpha$. Recall that there exist $\phi,\varphi\in$Bij$(\Gamma_0)$  such that $g+\phi(g)+\varphi(g)=g_0^0$  for every $g\in\Gamma_0$ ($g_0^0$ is the identity element of $\Gamma_0$) by Lemma \ref{bijection}.  Let  $f\colon\zet_{|\Gamma_0|}\rightarrow\Gamma_0$ be a bijection. Define $A_i^j$ by replacing an element $a_{i,h}$ in $\Lambda$ by $a_{i,1}^j=(\phi(f(j)),a_{i,1})$, $a_{i,2}^j=(\varphi(f(j)),a_{i,2})$, $a_{i,h}^j=(\gamma^{-1}(f(j)),a_{i,h})$ for $j=0,1,\ldots,|\Gamma_0|-1$, $i=1,2,\ldots,t$, $h=3,4,\ldots,2m+1$. Observe that $\sum_{h=3}^{2m+1}a_{i,h}^j=(f(j),a_0)$ for $j=0,1,\ldots,|\Gamma_0|-1$, $i=1,2,\ldots,t$.  Therefore  $A_1^0,\ldots,A_t^0,A_1^1,\ldots,A_t^1,\ldots,A_1^{|\Gamma_0|-1},$ $\ldots,A_t^{|\Gamma_0|-1}$ is a partition of the group $\Gamma$ such that for every $1 \leq i \leq t$, $0\leq j\leq|\Gamma_0|-1$ we have $|A_i^j| = 2m+1$ with $\sum_{a\in A_i^j} a =(g_0^0,a_0)=g_0$.\end{proof}

By above Lemmas~\ref{mZSPeven} and \ref{mZSPodd} the following theorem is true.
\begin{theorem}\label{mZSP} An Abelian group $\Gamma$  of order  $n$  has the $m$-ZSP-property  if and only if  $m\geq3$, $m|n$ and $\Gamma\in\gr$.
\end{theorem}
Since any group $\Gamma\in\gr$ has the  $m$-ZSP property for any $m\geq3$,  we state the following conjecture.
\begin{conjecture}Let $\Gamma\in\gr$ be of an even order $n$. For
every partition $n-1 = r_1 + r_2 + \cdots + r_t$ of $n-1$, with $r_i \geq 3$ for $1 \leq i \leq t$ and for any possible positive integer $t$, there is a partition of $\Gamma-\{g_0\}$ into pairwise disjoint subsets $A_1, A_2,\ldots, A_t$, such that $|A_i| = r_i$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$. 
\end{conjecture}
Recently, it was proved that the conjecture is true for $t=3$, see \cite{Cic3}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Group distance labeling of odd regular graphs}\label{GDM}

The \textit{lexicographic product} or \textit{graph composition} $G \circ H$ of graphs $G$ and $H$ is a graph such that  the vertex set of $G \circ H$ is the Cartesian product $V(G) \times V(H)$; and
    any two vertices $(u,v)$ and $(x,y)$ are adjacent in $G \circ H$ if and only if either $u$ is adjacent with $x$ in $G$ or $u = x$ and $v$ is adjacent with $y$ in $H$. Note that $G\circ H$ and $H\circ G$ are not isomorphic in general. One can imagine obtaining $G\circ H$ by blowing up each vertex of $G$ into a copy of $H$.


%It is interesting that if you replace each vertex of an $r$-regular graph by some specific $p$-regular graph (\cite{ref_AnhCicPetTep1}), then the resulting graph is group distance magic. 
By results proved in \cite{ref_AnhCicPetTep1} (see Theorem 2.2, \cite{ref_AnhCicPetTep1}) we have the following  observation.
\begin{observation}[\cite{ref_AnhCicPetTep1}]\label{Iztok} If $G$ is a regular graph, then the lexicographic product $G\circ \overline{K}_{2n}$  is group distance magic for any $n\geq1$.
\end{observation}


When $G$ is an arbitrary graph (not necessarily regular), we can use the $m$-ZSP-property of $\Gamma\in\gr$ to obtain the following.


\begin{theorem}\label{kn} If $G$ is a graph of order $t$, then the lexicographic product $G\circ \overline{K}_{n}$ has a $\Gamma$-distance magic labeling for $n\geq3$ and any $\Gamma\in\gr$ of order $nt$.
\end{theorem}
\begin{proof}
Let $G$ be a  graph with the vertex set $V(G) = \{v_0,v_1, \ldots, v_{t-1} \}$, and let  $V(\overline{K}_{2n+1}) = \{x_0,x_1,\ldots, x_{n-1}\}$.  Since $\Gamma\in\gr$,  there exists a partition $A_1,A_2,\ldots,A_t$ of  $\Gamma$ such that   $|A_i| =n$ with $\sum_{a\in A_i} a =g_0$ for every $1 \leq i \leq t$ by Theorem~\ref{mZSP}.  Label the vertices of the  $i$-th copy of $\overline{K}_{n}$ using elements from the set $A_i$ for $i=1,2,\ldots,t$. Therefore $w(v_i,x_j)=g_0$.
\end{proof}
Theorem~\ref{kn} implies immediately the following observation.
\begin{observation}\label{knob} If $G$ is a graph of odd order $t$, then the lexicographic product $G\circ \overline{K}_{2n+1}$ is  group distance magic for $n\geq1$.
\end{observation}


Observe that if $G$ is an odd regular graph, then  the lexicographic product $G\circ \overline{K}_{2n+1}$ is also an odd regular graph. Thus the order of the graph $G\circ \overline{K}_{2n+1}$ is even by handshaking lemma. Recall that $K_{3,3,3,3}\cong K_4\circ\overline{K}_{3}$ which is $\zet_2\times\zet_2\times\zet_3$-distance magic \cite{CicFro}. From Theorems  \ref{gr:odd}  and  \ref{kn} we obtain the following theorem showing that there exist infinitely many odd regular  $\Gamma$-distance magic graphs where $\Gamma$ is a group with more than one involution. 
\begin{theorem}If $G$ is an odd regular graph of order $t$, then the lexicographic product $G\circ \overline{K}_{2n+1}$ has a $\Gamma$-distance magic labeling for $n\geq1$ and $|\Gamma|=(2n+1)t$ if and only if $\Gamma$ has more than one involution.
\end{theorem}




For bipartite Eulerian graphs (i.e., a graph with all vertices of even degrees)  such that the partition sets have the same cardinality, we have the following theorem.
\begin{theorem} If $G$ is an Eulerian bipartite graph of order $t\equiv 2 \pmod 4$ with partition sets of the same cardinality, then $G\circ \overline{K}_{2n+1}$  is group distance magic for $n\geq1$. \label{dwudzielne}
\end{theorem}

\begin{proof}%[Proof of Theorem~\ref{Thm:FabGraphs}]
Notice that  the partition sets $V_0$, $V_1$ of $G$ have the same cardinality $k=t/2$.
Let  $V_0 = \{v_0,v_1, \ldots, v_{k-1} \}$, $V_1 = \{w_0,w_1, \ldots, w_{k-1} \}$ whereas  $V(\overline{K}_{2n+1}) = \{x_0,x_1,\ldots, x_{2n}\}$.
Let $\Gamma$ be any Abelian group of order $(2n+1)t$. Observe that $\Gamma\cong \zet_2\times \Lambda$ for some Abelian group $\Lambda$ of odd order $(2n+1)t/2$ because $t\equiv 2 \pmod 4$. \\
Since $2n+1\geq3$,  there exists a partition $A_1,A_2,\ldots,A_{k}$ of the group $\Lambda$ such that for every $1 \leq i \leq k$, $|A_i| = 2n+1$ with $\sum_{a\in A_i} a =a_0$ by Theorem~\ref{mZSP}. Define $A_i^0$ by replacing each element $a$ in $\Lambda$ by $(0,a)$  and analogously $A_i^1$ by replacing each element $a$ in $\Lambda$ by $(1,a)$. Label the vertices of the  $i$-th copy of $\overline{K}_{2n+1}$ in $V_0$, $V_1$ using elements from the set $A_i^0$, $A_i^1$, respectively for $i=1,2,\ldots,k$.

Notice that $w(x)=g_0$ for any $x\in V( G\circ \overline{K}_{2n+1})$ because $x$ has an even degree.\end{proof}




Recall that any regular  bipartite graph $G$ has the partition sets with the same cardinality, therefore by above Observation~\ref{Iztok} and Theorem~\ref{dwudzielne} we easily obtain the following observation. 

\begin{observation} If $G$ is an $r$-regular bipartite   graph of order $t\equiv 2 \pmod 4$ and $n\geq2$, then $G\circ \overline{K}_{n}$  is group distance magic if and only if $rn$ is even.
\end{observation}
\begin{proof} Assume first that $rn$ is odd, then $nt\equiv 2 \pmod 4$ and by Theorem~\ref{2mod4_1} there does not exist a group $\Gamma$ such that $G\circ \overline{K}_{n}$ admits a $\Gamma$-distance labeling.
If $n$ is even then we are done by Observation~\ref{Iztok}. In the last case $n$ odd and $r$ even we apply Theorem~\ref{dwudzielne}.\end{proof}

 %Hence  $\Gamma\cong \zet_2\times \Lambda$ for some Abelian group $\Lambda$ of odd order $nt/2$. Therefore $\Gamma$ has exactly one involution and


\section{$\Gamma$-Kotzig arrays}\label{Kotzig}
In \cite{Wal} Marr and Wallis give a definition of a \textit{Kotzig array} as a $j\times k$ grid, where each row is a permutation of $\{0,1,\dots,k-1\}$ and each column has the same sum. Note that a  Latin square  is a special case of a Kotzig array.
 \begin{lemma}[\cite{Wal}]\label{array}
A Kotzig array of size $j \times k$ exists if and only if $j>1$ and $j(k-1)$ is even.
\end{lemma}




Kotzig arrays play important role in graph labelings.  There are many constructions based on Kotzig arrays of various magic-type labelings of copies and products of regular graphs \cite{ref_CicFroSin,ref_InaLlaMor,Wal}. 

For an Abelian group $\Gamma$ of order $k$ we define a $\Gamma$-\textit{Kotzig array}  of size $j\times k$ as a  $j\times k$ grid, where each row is a permutation of elements of $\Gamma$ and each column has the same sum. Denote by $x_{i,j}$ the entry in the $i$-th row and $j$-th column of
the array.
\begin{lemma}
A $\Gamma$-Kotzig array of size $2 \times k$ exists for any Abelian group $\Gamma$.
\end{lemma}
\begin{proof} Let $\Gamma=\{g_0,g_1,\ldots,g_{k-1}\}$. Let $x_{1,i}=g_i$, $x_{2,i}=-g_i$ for $i=0,1,\ldots,k-1$.\end{proof}

\begin{lemma}
A $\Gamma$-Kotzig array of size $3 \times k$ exists for any Abelian group $\Gamma\in\gr$.
\end{lemma}
\begin{proof} Let $\Gamma=\{g_0,g_1,\ldots,g_{k-1}\}$.  There exist $\phi,\varphi\in$Bij$(\Gamma)$  such that $g+\phi(g)+\varphi(g)=g_0$  for every $g\in\Gamma$ by Lemma~\ref{bijections}. Let $x_{1,i}=g_i$, $x_{2,i}=\phi(g_i)$, $x_{2,i}=\varphi(g_i)$.\end{proof}


\begin{theorem}
A $\Gamma$-Kotzig array of size $j \times k$ exists if and only if
$j>1$ and $j$ is even or $\Gamma\in \gr$.
\end{theorem}
\begin{proof} Obviously for $j=1$ there does not exist a $\Gamma$-Kotzig array. 
 Assume first $j$ is even. To construct
a $\Gamma$-Kotzig array of size $j\times k$, we simply take $j/2$ of $\Gamma$-Kotzig arrays of size $2\times k$ and ``glue'' them into an array $j\times k$.


Suppose $j$ is odd and there exists a $\Gamma$-Kotzig array for a group $\Gamma$ containing only one involution $\iota$.
Recall that $kg=g_0$ for any $g\in \Gamma$ and  for some $\mu\in \Gamma$ we have $x_{1,i}+x_{2,i}+\ldots+x_{j,i}=\mu$ for any $i=1,2,\ldots,k$.  On one hand $\sum_{i=1}^k\sum_{l=1}^jx_{l,i}=k\mu=g_0$.  On the other $\sum_{i=1}^k\sum_{l=1}^jx_{l,i}=\sum_{l=1}^j\sum_{i=1}^kx_{l,i}=j\sum_{g\in \Gamma}g=j\iota=\iota$, a contradiction.

Assume now that $\Gamma\in \gr$. To construct a $\Gamma$-Kotzig array of size $j\times k$, we simply take $(j-3)/2$ of $\Gamma$-Kotzig arrays of size $2\times k$, one $\Gamma$-Kotzig array of size $3\times k$ and ``glue'' them into an array $j\times k$.\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Acknowledgements}
%Thanks to Professor Querty for suggesting the proof of
%Lemma~\ref{lem:Technical}.



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