On the chromatic polynomial of a graph

Abstract.

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, $\frac\Box \lambda-n+\alpha\Box \Box \lambda\Box$ ($\frac\Box \lambda-n+\alpha-1\Box \Box \lambda-n+\alpha\Box$)^α≤$\frac\Box P(G,\lambda-1)\Box \Box P(G,\lambda)\Box$≤$\frac\Box \lambda-\omega\Box \Box \lambda\Box$ ($\frac\Box \lambda-1\Box \Box \lambda\Box$)^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.¶These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)($\frac\Box n-1\Box \Box n\Box$)^{n −ω}≤μ(G)≤n−α($\frac\Box \alpha-1\Box \Box \alpha\Box$) ^α.