Two Chromatic Polynomial Conjectures

LetP(λ) be the chromatic polynomial of a graph. We show thatP(5)−1 P(6)2 P(7)−1can be arbitrarily small, disproving a conjecture of Welsh (and of Brenti, independently) thatP(λ)2⩾P(λ−1) P(λ+1)and also disproving several other conjectures of Brenti. Secondly, we prove that if the graph has n vertices, thenP(n) P(n−1)−1⩾2.718253,approaching a conjecture of Bartels and Welsh thatP(n) P(n−1)−1⩾e(eis 2.718281 …).