For any positive integer n, let L_n denote the set of simple graphs of order n. For any graph G in L_n, let P(G,λ) denote its chromatic polynomial. In this paper, we first show that if G ∈ L_n and X(G) ≤ n-3, then P(G,λ) is zero-free in the interval (n-4+β/6-2/β,+∞), where β = (108 + 12(93)~(1/2)~(1/3) and β/6-2/β (=0.682327804…) is the only real root of x~3+x-1; we proceed to prove that whenever n-6 ≤ X(G) ≤ n-2, P(G,λ) is zero-free in the interval (「n+X(G))/2」-2,+∞). Some related conjectures are also proposed.
展开▼
