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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