Proof of a Chromatic Polynomial Conjecture

In this paper, all graphs considered are simple graphs. We always suppose that G is a graph. Let V(G), E(G), v(G), and e(G) be the vertex set, edge set, order and number of edges of G. For a positive integer *, a *-colouring of G is a mapping f : V(G) [1, ..., *] such that f (x){ f ( y) whenever x and y are adjacent in G. Let P(G, *) denote the number of *-colourings in G. It is well known that P(G, *) is a polynomial in *, called the chromatic polynomial of G. In this paper, P(G, *) is considered to be a polynomial in the real number *. We are concerned with the following conjecture about P(G, *),