The Chromatic Polynomial of $P_2 \times P_n$ and $C_3 \times P_n$

Abstract

We provide formulas for the chromatic polynomials of the Cartesian product of a path on two vertices with a path on $n$ vertices, and a cycle on three vertices with a path on $n$ vertices. The result for the Cartesian product of paths involves the trinomial coefficients. We define a new sequence of numbers, $CP(n,i)$, which are part of the result for the chromatic polynomial of the Cartesian product of the three cycle and the path. In both cases the chromatic polynomial is factored in the form $\lambda (\lambda -1)$.