Two chromatic polynomial conjectures
Let \(P(t)\) be the chromatic polynomial of a graph. It is shown that \(P(5)^{-1}P(6)^2 P(7)^{-1}\) can be arbitrarily small, disproving a conjecture of Welsh that \(P(t)^2\geq P(t- 1)P(t+1)\), and also disproving several other conjectures of Brenti. Secondly, it is proved that if the graph has \(n\) vertices, then \(P(n)P(n- 1)^{-1}\geq 2.718253\), approaching a conjecture of Welsh and Brenti that \(P(n)P(n-1)^{-1}\geq e\).
scientific article; zbMATH DE number 1076878