2022/09/21 by Fiorenza Morini, Morini, Fiorenza, Marco Antonio Pellegrini +1 · 1 citation
Computer Science · Engineering · #05B15 #05B30 #05C78 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2209.10246
openalex publication_date 2022/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce a special class of partially filled arrays. A magic partially filled array MPFΩ(m,n; s,k) on a subset Ω of an abelian group (Γ,+) is a partially filled array of size m× n with entries in Ω such that (i) every ω∈ Ω appears once in the array; (ii) each row contains s filled cells and each column contains k filled cells; (iii) there exist (not necessarily distinct) elements x,y∈ Γ such that the sum of the elements in each row is x and the sum of the elements in each column is y. In particular, if x=y=0Γ, we have a zero-sum magic partially filled array 0MPFΩ(m,n; s,k). Examples of these objects are magic rectangles, Γ-magic rectangles, signed magic arrays, (integer or non integer) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, i.e., of an MPFΩ(m,n;s,k) where Ω=\1,2,…,nk\⊂ℤ. We also construct zero-sum magic partially filled arrays when Ω is the abelian group Γ or the set of its nonzero elements.