Zero-sum partitions of Abelian groups of order $2^n$ - Sylwia Cichacz; Karol Suchan (Discrete Mathematics & Theoretical Computer Science)
The following problem has been known since the 80's. Let $Γ$ be an Abelian group of order $m$ (denoted $|Γ|=m$), and let $t$ and $m_i$, $1 ≤ i ≤ t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $Γ^*=Γ\setminus\{0\}$, the set of non-zero elements of $Γ$, can be partitioned into disjoint subsets $S_i$, $1 ≤ i ≤ t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 ≤ i ≤ t$. It is easy to check that $m_i≥ 2$ (for every $i$, $1 ≤ i ≤ t$) and $|I(Γ)|≠ 1$ are necessary conditions for the existence of such partitions, where $I(Γ)$ is the set of involutions of $Γ$. It was proved that the condition $m_i≥ 2$ is sufficient if and only if $|I(Γ)|\in\{0,3\}$. For other groups (i.e., for which $|I(Γ)|≠ 3$ and $|I(Γ)|>1$), only the case of any group $Γ$ with $Γ\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i≥ 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|Γ|$ is large enough and $|I(Γ)|>1$, then $m_i≥ 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i≥ 3$ is sufficient for $Γ$ such that $|I(Γ)|>1$ and $|Γ|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.