On upper bounds for real roots of chromatic polynomials Original Research Article

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,λ) denote its chromatic polynomial. In this paper, we first show that if G∈Gn and χ(G)⩽n−3, then P(G,λ) is zero-free in the interval (n−4+β/6−2/β,+∞), where β=(108+1293)1/3 and β/6−2/β (=0.682327804…) is the only real root of x3+x−1; we proceed to prove that whenever n−6⩽χ(G)⩽n−2, P(G,λ) is zero-free in the interval (⌈(n+χ(G))/2⌉−2,+∞). Some related conjectures are also proposed.