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Balanced equi-n-squares

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.37236/9118
  3. Publisher: Australian National University , 2020
  4. Abstract:
  5. We define a d-balanced equi-n-square L = (lij ), for some divisor d of n, as an n×n matrix containing symbols from Zn in which any symbol that occurs in a row or column, occurs exactly d times in that row or column. We show how to construct a d-balanced equi-n-square from a partition of a Latin square of order n into d×(n/d) subrectangles. In design theory, L is equivalent to a decomposition of Kn,n into d-regular spanning subgraphs of Kn/d,n/d . We also study when L is diagonally cyclic, defined as when l(i+1)(j+1) = lij + 1 for all i, j ∈ Zn, which corresponds to cyclic such decompositions of Kn,n (and thus α-labellings). We identify necessary conditions for the existence of (a) d-balanced equi-n-squares, (b) diagonally cyclic d-balanced equi-n-squares, and (c) Latin squares of order n which partition into d × (n/d) subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed d ≥ 1 when n is sufficiently large, and we resolve the existence problem completely when d ∈ {1, 2, 3}. Along the way, we identify a bijection between α-labellings of d-regular bipartite graphs and what we call d-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either d or 0 filled cells in each row and column. We use d-starters to construct diagonally cyclic d-balanced equi-n-squares, but this also gives new constructions of α-labellings. © The authors. Released under the CC BY-ND license (International 4.0)
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  8. Source: Electronic Journal of Combinatorics ; Volume 27, Issue 4 , 2020 , Pages 1-35
  9. URL: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i4p8