Let G be a simple graph Prove that the chromatic polynomial

.Let G be a simple graph. Prove that the chromatic polynomial PG(k) is the product of the chromatic polynomials of its components.

Solution

For any shading of G the nonempty shading classes constitute a segment of V (G) where each part is a steady vertex set. We may tally those colorings that give a specific parcel and include them up for every such segment to nd the aggregate number of colorings. Since V (G) is a nite set, it has a nite number of allotments, so it is sucient to demonstrate that the quantity of colorings for a solitary segment is a polynomial of k.

Settle a parcel with p parts, each of them being a steady set. By doling out a dierent shading to each part, we get every one of the colorings having a place with the segment. We may pick the rst shading in k conceivable ways, the second in k1 ways, and so on so there are k(k1)...(kp+1) colorings, which is clearly a polynomial. Take note of this additionally works when k < p