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) is zero-free in the interval (1, β), where β = 2 + 1 6 3 √ 12 √ 93− 108− 1 6 3 √ 12 √ 93 + 108 = 1.317672196 · · · , and P (G, β) = 0 for some graph G with domination number 2. We also show that if ∆(G) ≥ v(G) − 2, then P (G,λ) is zero-free in the interval (1, β′), where β′ = 5 3 + 1 6 3 √ 12 √ 69− 44− 1 6 3 √ 12 √ 69 + 44 = 1.430159709 · · · , and P (G, β′) = 0 for some graph G with ∆(G) = v(G)− 2. ∗Supported by AcRf project RI 5/06 DFM of National Institute of Education, Singapore. †Corresponding author. Email:[email protected].