Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set
Sylwia Cichacz
TL;DR
The paper investigates partitioning finite Abelian groups into zero-sum blocks via complete mappings and applies these partitions to the existence of Gamma-magic rectangle sets $MRS_{\Gamma}(a,b;c)$. It proves that, for $k>1$, $c$ odd, and $\exp(\Gamma)\not\equiv 0\pmod{8}$, such a set exists precisely when $a$ and $b$ are both even or $\Gamma\in\mathcal{G}$ and $\{a,b\}\neq\{2k+1,2\}$, using starter constructions and blow-up lemmas together with Kotzig-array methods. The approach leverages zero-sum partitions induced by complete mappings to build $MRS_{\Gamma}$ in challenging parity and structural cases, advancing toward a complete characterization in this family of problems. This work provides new sufficient conditions and a framework that clarifies the role of group structure, especially the exponent modulo 8, in the existence of $\Gamma$-magic rectangle sets with mixed odd/even dimensions.
Abstract
A complete mapping of a group $Γ$ is a bijection $\varphi\colon Γ\to Γ$ for which the mapping $x \mapsto x+\varphi(x)$ is a bijection. In this paper we consider the existence of a complete mapping $\varphi$ of $Γ$ and a partition $S_1,S_2,\ldots S_t$ of elements of $Γ$, such that $\sum_{s\in S_i}s=\sum_{s\in S_i}\varphi(s)=0$ for every $i$, $1 \leq i \leq t$. A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$. While a complete characterization of MRS$_Γ(a,b;c)$ exists for cases where $\{a,b\}\not=\{2k+1,2^α\}$, the scenario where $\{a,b\}=\{2k+1,2^α\}$ remains unsolved for $α>1$. Using the partition of $Γ$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $Γ$-magic rectangle set MRS$_Γ(2k+1, 2^α;c)$ exists.