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)[les ]3 and δ(G)>n/3 is of Fd-type for some d[ges ]1, where Fd 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)[ges ]4 and δ(G)>n/3 contains the Mycielski graph (see Figure 2) as a subgraph.