Proof of a chromatic polynomial conjecture (Q1569069)

Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\) (i.e., the number of mappings \(f\) from the vertex set of \(G\) to \(\{1,2,\dots, \lambda\}\) such that \(f(x)\neq f(y)\) whenever \(x\) and \(y\) are adjacent vertices in \(G\) if \(\lambda\) is a positive integer). We prove a conjecture on chromatic polynomials proposed by \textit{J. E. Bartels} and \textit{D. J. A. Welsh} [Lect. Notes Comp. Sci. 920, 373-387 (1995)]: \(P(G, n)(P(G, n-1))^{-1}\geq n^n/(n- 1)^n> e\), where \(n\) is the number of vertices in \(G\) and \(e\) is the base of the natural logarithm.