On Prime Matrix Product Factorizations

TL;DR

This work provides a complete characterization of prime graphs under matrix-product factorizations, showing that $G$ is prime iff every factorization $G=HK$ contains a matched-pair endpoint in each component. The authors develop a combinatorial framework centered on diamond conditions and matched pairs, yielding structural results such as that a factorable forest must pair isomorphic components and, for square-free graphs, a factorization forces a perfect-matching factor. They apply these insights to derive concrete classifications: grids $P_n\square P_m$ are factorable precisely when $m,n$ are even, while torus grids $C_m\square C_n$ are factorable for all $m,n$, with explicit constructions in each regime. These contributions solve open questions from prior work, including aspects of grids and torus factorizations, and lay groundwork for broader classifications and algorithmic approaches to matrix-product factorizations in graphs.

Abstract

A graph $G$ factors into graphs $H$ and $K$ via a matrix product if $A = BC$, where $A$, $B$, and $C$ are the adjacency matrices of $G$, $H$, and $K$, respectively. The graph $G$ is prime if, in every such factorization, one of the factors is a perfect matching that is, it corresponds to a permutation matrix. We characterize all prime graphs, then using this result we classify all factorable forests, answering a question of Akbari et al. [\emph{Linear Algebra and its Applications} (2025)]. We prove that every torus is factorable, and we characterize all possible factorizations of grids, addressing two questions posed by Maghsoudi et al. [\emph{Journal of Algebraic Combinatorics} (2025)].

Figures (4)

• Figure 1: $P_8 \cup P_8$ is factored into $P_8 \cup P_8$ and a perfect matching.
• Figure 2: The left graph depicts $P_2 \square P_4$, which is factored into the two graphs on the right.
• Figure 3: Factorization of $C_3\square C_3$ into two 2-regular graphs.
• Figure 4:

Theorems & Definitions (61)

• Definition 1.1
• Theorem 1.1
• Theorem 1.1
• Example 1
• Theorem 1.1
• Example 2
• Theorem 1.1
• Example 3
• Definition 2.2
• Definition 2.3