A note on the shameful conjecture

Let $P_G\left(q\right)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, $\frac{P_G\left(n\right)}{P_G(n-1)}\ge \frac{n^n}{{(n-1)}^n}\text{.}$ Let $\mu \left(G\right)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\mu \left(G\right)\ge \mu \left(O_n\right)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F.M. Dong, who in fact showed that, $\frac{P_G\left(q\right)}{P_G(q-1)}\ge \frac{q^n}{{(q-1)}^n}$ for all $q\ge n$. There are examples showing that this inequality is not true for all $q\ge 2$. In this paper, we show that the above inequality holds for all $q\ge 36D^{3/2}$, where $D$ is the largest degree of $G$. It is also shown that the above inequality holds true for all $q\ge 2$ when $G$ is a claw-free graph.