Graphs which satisfy $\chi(G) > \chi(H)$ and $p_G(x) > p_H(x)$? I am trying to find graphs $G$ and $H$ which have the same number of vertices and edges, but which satisfy $\chi(G) > \chi(H)$ and $p_G(x) > p_H(x)$ for sufficiently large $x$, where $p_G(x)$ is the chromatic polynomial of $G$, and $\chi(G)$ is the chromatic number of $G$.
All the examples I have come up with so far where $\chi(G) > \chi(H)$ lead to $p_G(x) < p_H(x)$ for sufficiently large $x$; for example, the union of $C_3$ and a single disconnected vertex has $\chi(G) = 3$ and $p_G(x) = x[(x-1)^3 - (x-1)]$, while the path on 4 vertices $P_4$ has $\chi(H) = 2$ and $p_H(x) = x(x-1)^3$, since it is a tree; this leads to $p_G(x) < p_H(x)$ for sufficiently large $x$, which is not what I want. I have tried many other constructions involving wheels and trees as well, but they all run into the same problem, so any suggestions would be greatly appreciated.
 A: We know that the coefficients of the chromatic polynomial have alternating signs, so let's write $$p_G(x) = \sum_{i=1}^n (-1)^{n-i} a_i x^i$$
If we denote $p_H(x) = \sum_{i=1}^n (-1)^{n-i} b_i x^i $, then we can express the difference as
$$ p_G(x) - p_H(x) = \sum_{i=1}^n (-1)^{n-i} (a_i-b_i) x^i $$
So whether $p_G(x) > p_H(x)$ for large $x$ will depend on the sign of the largest power of $x$ in $p_G - p_H$. We want then to see which is the first $i$ such that $a_i \neq b_i$.
Since $G$ and $H$ both have $n$ vertices and $m$ edges, then we know that $a_n=b_n=1$ and $a_{n-1}=b_{n-1}=m$. So the $n$ and $n-1$ powers always cancel out. We now want to use a non-trivial result on the high-order coefficients of $p_G$, which can be found in Dong, Kong, Teo, Chromatic polynomials and chromaticity of graphs. If $g=g(G)$ is the girth of $G$, then
$$ a_{n-i} = \binom{m}{i} \quad i=1\dots g-2$$
$$ a_{n-g+1} = \binom{m}{g-1} - n_G(C_g) $$
where $n_G(C_g)$ if the number of copies of $C_g$ in $G$. 
This sounds terribly complicated, but it is not: what this implies is that, if $G$ and $H$ have different girth or have the same girth but with a different "multiplicity", then the maximum degree of $p_G-p_H$ is $x^{n-g+1}$, where $g$ is the minimum of the girth of $G$ and of $H$. We cannot say much for the other cases.
In conclusion, if $g(G)>g(H)$ then 
$$ \mathrm{sign}(p_G(x) - p_H(x)) \sim (-1)^{n-g(H)}, $$
while if $g(G)< g(H)$ then
$$ \mathrm{sign}(p_G(x) - p_H(x)) \sim (-1)^{n-g(G)+1}.$$
The fact that the sign is not just depending on which one of the graphs has a smallest girth, but also on the parity of the girth and of $n$, makes finding an example of what you want a bit harder. This is what I managed to come up with, but I suspect there should be smaller examples.
The example
Take $G$ to be the Grötzsch graph with 10 extra isolated vertices. It has 21 vertices and 20 edges. Its chromatic number is 4 and its girth is also 4.
Take $H$ to be the path graph with 21 vertices (it has 20 edges). Its chromatic number is 2 (it is a bipartite graph) and its girth is infinity (it contains no loops).
Then $\chi(G) > \chi(H)$, but from the argument above $\mathrm{sign}(p_G(x) - p_H(x))$ should be $(-1)^{21-4+1}=+1$ for large $x$. Indeed with the help of a computer we can see that
$$ p_G(x) - p_H(x) = 10x^{18} + \text{smaller order terms},$$
so clearly $p_G(x)>p_H(x)$ for a large enough $x$. 
