Proof of a Chromatic Polynomial Conjecture

Let P(G, λ) denote the chromatic polynomial of a graph G. It is proved in this paper that for every connected graph G of order n and real number λ⩾n, (λ−2)n−1 P(G, λ)−λ(λ−1)n−2 P(G, λ−1)⩾0. By this result, the following conjecture proposed by Bartels and Welsh is proved: P(G, n)(P(G, n−1))−1>e for every graph G of order n.