I was playing around with the chromatic polynomial (denoted here by $\chi\_G(x)$) and I have made the following conjecture.


Let $(G\_n)\_{n \ge 1}$ be a sequence of graphs with $v(G\_n) \to \infty$ ($v(G\_n)$ denotes the number of vertices of $G\_n$) and $e(G\_n) \to \infty$ ($e(G\_n)$ denotes the number of edges of $G\_n$).


For each $x \neq 0$, let us define the following transform of the chromatic polynomial of $G\_n$
$$
 \psi\_{G\_n}(x) = \frac{x^{v(G\_n)}}{e(G\_n)^{v(G\_n)}} \chi\_{G\_n}\left( \frac{e(G\_n)}{x} \right).
$$


The conjecture is that for each fixed real number $x \neq 0$, we have $\psi\_{G\_n}(x) \to \exp(-x)$ as $n$ goes to infinity.


I have checked the conjecture for a few sequences of graphs: for example, $G\_n$ being the complete graph $K\_n$, for $G\_n$ being a tree on $n$ vertices and for $G\_n$ being a collection of $n$ independent edges (a matching on $2n$ vertices).


Does anyone know if this is well-known?


PS: I am not sure if the conditions on $v(G\_n)$ and $e(G\_n)$ are the right one. Any comments on this are welcome as well.



On a limit involving a transform of the chromatic polynomial
Here is a heuristic argument which perhaps someone can make
rigorous. I write $v\_n=v(G\_n)$ and $e\_n=e(G\_n)$. Let
$$ \chi\_{G\_n}(x) =
 x^{v\_n}-c\_{n,v\_n-1} x^{v\_n-1}+c\_{n,v\_n-2}x^{v\_n-2}-\cdots. $$
I claim that for fixed $k\geq 0$,
$$ \lim\_{n\to\infty} \frac{c\_{n,v\_n-k}}{e\_n^k} = \frac{1}{k!}. $$
One can prove this by noting that (by the Broken Circuit Theorem, for
instance, which shows that $c\_{n,v\_n-k}$ increases as we add more
edges to $G\_n$) $c\_{n,v\_n-k}$ is bounded below by its value when $G\_n$
is a tree, and is bounded above by its value when $G\_n$ is a complete
graph. The claimed result is easily verified for trees and complete
graphs (in the latter case, using known asymptotics for the Stirling
numbers of the first kind). Perhaps there is a more direct proof, but
in any case, if we don't worry about justifying interchanging limits
and sums, we get
$$ \lim\_{n\to\infty} \frac{x^{v\_n}}{e\_n^{v\_n}}\chi\_{G\_n}\left(
 \frac{e\_n}{x}\right) = \sum\_{k\geq 0} \lim\_{n\to\infty}
 \frac{(-1)^k c\_{n,v\_n-k}x^k}{e\_n^k} $$
$$ \qquad = \sum\_{k\geq 0} \frac{(-1)^k x^k}{k!} = \exp(-x). $$


