factor

General

The factors of a natural number \(n\) are those whole numbers which divide it evenly. Since every number is divisible by itself and 1, neither 1 nor \(n\) is considered a proper factor of \(n\).

More generally, if an expression can be written as a product of other expressions, then those other expressions are its factors. For instance, a polynomial with rational coefficients may always be factored into a product of first and/or second degree polynomial factors.

A number (or other expression) with no proper factors is called prime.

Graph Theory

A factor of a graph is a spanning subgraph having at least one edge. In many contexts, it is interesting to determine whether some graph \(G\) can be decomposed into the edge-disjoint union of factors with some prescribed property. Such a decomposition is called a factorization of \(G\). Often, the property in question is regularity of degree \(k\). In this case, the factors are called \(k\)-factors, and the factorization a \(k\)-factorization. If \(G\) has a \(k\)-factorization, it is called \(k\)-factorable.