How could the following two definitions of chromatic polynomials be the same?
The chromatic polynomial is the polynomial $P(G,x)$ with variable $x$ that gives the number of ways of coloring the graph $G$ with $x$ or fewer colors.
and
The chromatic polynomial is the polynomial $P(G,x)$ with variable $x$ that gives the number of ways of coloring the graph $G$ with exactly $x$ colors.
I see that the number of colorings is the same in both the definitions upto $x=k$, where $k$ is the chromatic number of $G$. But, if $x>k$, I quite dont see how coloring the graph with $\textit{exactly}$ $x$ colors is the same as coloring it with $x$ or fewer colors. Thanks beforehand.
Best Answer
Your first definition is the correct one, I've never seen the term "chromatic polynomial" with the second definition.
The second definition is false, and is an incomplete form for chromatic polynomials.