Structure and Decomposition of Deltoids in Abelian Groups
Mohsen Aliabadi, Jozsef Losonczy
TL;DR
The paper develops a deltoid-based framework to study defective (partial) matchings between finite subsets $A,B$ of an abelian group $G$ with $|A|=|B|$. By combining Hall–Ore type deficiency criteria with Dyson's $e$-transform, it yields a sharp characterization of when a partial matching with defect $d$ exists and provides an explicit formula for the deficiency $\delta(A,B)$. A structure theorem identifies obstructions to small-defect matchings via coset decompositions, and the authors introduce partition numbers $\rho(A,B)$ and $\lambda(A,B)$ to analyze how $A$ and $B$ can be decomposed into left- and right-admissible pieces, including precise necessary and sufficient conditions and minimality results. The work also connects to defective Chowla-type sets and discusses extensions to non-abelian settings, while outlining open problems regarding the left-partition theory and extremal counts of pairs with fixed defect.
Abstract
Deltoids provide a natural framework for studying defective (partial) matchings in abelian groups, and we develop both structure and existence results in this setting. Given finite subsets $A$ and $B$ of an abelian group $G$, a matching is a bijection $f:A\to B$ such that $af(a)\notin A$ for all $a\in A$, a definition motivated by the study of canonical forms for symmetric tensors. We provide necessary and sufficient conditions for the existence of a partial matching with any prescribed defect, and then describe the minimal unavoidable defect for a pair $(A,B)$. We also define and examine a defective version of Chowla sets in the matching context. We prove a structure theorem identifying obstructions to the existence of partial matchings with small defect. Finally, within the deltoid setup, we establish max-min results on the partitioning of $A$ and $B$ into left- and right-admissible sets. Our tools mix results from transversal theory with ideas from additive number theory.