Factors of disconnected graphs and polynomials with nonnegative integer coefficients

TL;DR

This paper studies the unique factorization of disconnected finite graphs under various graph products, identifying conditions for uniqueness and explicitly describing non-unique cases for certain component counts.

Contribution

It provides new criteria for the uniqueness of graph factorization based on the number of components and characterizes all non-unique cases for specific component counts.

Findings

Unique factorization holds when the number of components is prime or in {1,4,8,9} under certain conditions.

Explicit descriptions of all non-unique factorizations for graphs with 6 or 10 components.

Results apply to Cartesian, strong, and direct graph products.

Abstract

We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has $n$ connected components, where $n$ is prime, or $n=1,4,8,9$, and satisfies some additional conditions, it factors uniquely under the given products. If, on the contrary, $n=6$ or 10, all cases of nonunique factorisation are described precisely.