On upper bounds for real roots of chromatic polynomials

For any positive integer n, let Gn denote the set of simple graphs of order n. For any graph G in Gn, let P(G; ) denote its chromatic polynomial. In this paper, we -rst show that if G ∈Gn and (G)6 n− 3, then P(G; ) is zero-free in the interval (n − 4 + =6 − 2= ;+∞), where = (108 + 12√93)1=3 and =6 − 2= (=0:682327804 : : :) is the only real root of x + x − 1; we proceed to prove that whenever n − 66 (G)6 n − 2, P(G; ) is zero-free in the interval ( (n+ (G))=2 − 2;+∞). Some related conjectures are also proposed. c © 2003 Elsevier B.V. All rights reserved.