Francesco Brenti, Expansions of Chromatic Polynomials and Log-Concavity, Transactions of the American Mathematical Society, Vol. 332, No. 2 (Aug., 1992), pp. 729-756 ...

Bhaswar B. Bhattacharya, COLLISION TIMES IN MULTICOLOR URN MODELS AND SEQUENTIAL GRAPH COLORING WITH APPLICATIONS TO DISCRETE LOGARITHMS, The Annals of Applied Probability, Vol. 26, No. 6 (December ...

The mathematical expression that gives the answer is called a chromatic polynomial. More complex chromatic polynomials can be written for more complex geometric objects.