Generalized Latin Square Graphs of Semigroups: A Counting Framework for Regularity

TL;DR

The paper introduces generalized Latin square graphs Γ(S) for finite semigroups, linking graph regularity to algebraic structure via a degree formula deg(v) = 2n - 3 + Q(i,j). Regularity occurs precisely when Q(i,j) is constant, a phenomenon termed compensated factorization, often enforced by congruence-induced block uniformity. By analyzing cancellative, band, inverse, and constant-image semigroups, the authors highlight how symmetry in the multiplication table drives regular GLSGs, including a non-cancellative order-4 example yielding a 9-regular graph. Computational enumeration up to order 6 using GAP/smallsemi indicates regular GLSGs are rare, with data and scripts made publicly available for reproducibility.

Abstract

We investigate the generalized Latin square graph $Γ(S)$ for a finite semigroup $S$, extending the combinatorial concept of Latin rectangle graphs. By analyzing factorization multiplicities $(N_S)$ and cancellation failure $(N_R,N_C)$, we derive the degree formula $°(v) = 2n - 3 + Q(i,j)$. We prove that graph regularity is equivalent to $Q(i,j)$ being constant -- a condition termed \emph{compensated factorization} -- which often arises from congruence-induced symmetries. A non-cancellative semigroup of order 4 generating a 9-regular graph is constructed. Finally, computational enumeration up to order $n = 6$ (using the \textsf{smallsemi} package for GAP) confirms that regularity is a rare property which decreases significantly as the order increases.

Figures (1)

• Figure 1: The generalized Latin square graph $\Gamma(S_1)$, forming a complete graph $K_3$ plus an isolated vertex.

Theorems & Definitions (52)

• Proposition 1.1
• proof
• Definition 3.1
• Example 1
• Example 2: A non-regular case
• Definition 4.1
• Remark 1
• Remark 2
• Corollary 4.2
• proof