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A graph G is called an H-type graph for some graph H if there is a mapping from V (G) to V (H) preserving edges. In this paper, we shall prove that: (1) every triangle-free graph G of order n with χ(G) 6 3 and δ(G) > n/3 is of F d-type for some d > 1, where F d is a certain d-regular triangle-free Hamiltonian Cayley graph of order 3d − 1, (2) every triangle-free graph G of order n with χ(G) > 4 and δ(G) > n/3 contains the Mycielski graph (see Figure 2) as a subgraph.

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In this paper, we shall prove that if the domination number of G is at most 2, then P (G, ) is zero-free in the interval (1, ), where = 2 + 1 6 3 12 √ 93 − 108 − 1 6 3 12 √ 93 + 108 = 1.317672196 . . . , and P (G, ) = 0 for some graph G with domination number 2. We also show that if (G) v(G) − 2, then P (G, ) is zero-free in the interval (1, ), where = 5 3 + 1 6 3 12 √ 69 − 44 − 1 6 3 12 √ 69 + 44 = 1.430159709 . . . , and P (G, ) = 0 for some graph G with (G) = v(G) − 2.