Latin squares with non-partitioning disjoint subsquares

Abstract

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an arbitrary number of subsquares, $k$. Requiring only that $h_1+\dots+h_k\leq n$ gives a variation of the problem that has few known results. In this paper we prove a general necessary condition for existence and completely determine existence when there are at most three subsquares or the subsquares are all of the same order. Importantly, we prove that if $h_1\geq h_2\geq\dots\geq h_k$ and $n\geq h_1+\sum_{i=1}^kh_i$ then such a latin square always exists.

Figures (5)

• Figure 1: A latin square of order 8 with disjoint subsquares
• Figure 2: An outline square associated to $(3^1 2^1 1^3)$.
• Figure 3: The outline square for an $\mathop{\mathrm{ILS}}\nolimits(n;h_1h_2)$
• Figure 4: An outline square for an $\mathop{\mathrm{ILS}}\nolimits(n;h_1h_2h_3)$
• Figure 5: Outline arrays for $F_\ell$ where $\ell\in[a-b]$

Theorems & Definitions (40)

• Theorem 1.1: evans1960embedding
• Theorem 1.2: heinrich2006latin
• Theorem 1.3: denes1963some
• Theorem 1.4: heinrich1982disjointkuhl2018latin
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Theorem 3 of hilton1980reconstruction
• Lemma 2.4
• Definition 2.5
• Definition 2.6