Domination numbers and zeros of chromatic polynomials

In this paper, we shall prove that if the domination number of G   is at most 2, then P(G,λ)P(G,λ) is zero-free in the interval (1,β)(1,β), whereβ=2+161293-1083-161293+1083=1.317672196…,and P(G,β)=0P(G,β)=0 for some graph G   with domination number 2. We also show that if Δ(G)⩾v(G)-2Δ(G)⩾v(G)-2, then P(G,λ)P(G,λ) is zero-free in the interval (1,β′)(1,β′), whereβ′=53+161269-443-161269+443=1.430159709…,and P(G,β′)=0P(G,β′)=0 for some graph G   with Δ(G)=v(G)-2Δ(G)=v(G)-2.