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.

The smallest minimal degree of an r-partite graph that guarantees the existence of a complete subgraph of order r has been found for the case r = 3 by Bollobás, Erdő and Szemerédi, who also gave bounds for the cases r ≥ 4. In this paper the exact value is established for the cases r = 4 and 5, and the bounds for r ≥ 6 are improved.

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Abstract. In this paper, we extend earlier results concerning the maximal number of induced complete r-partite graphs in a graph G of order n. In particular, we show that if t > 1 + log r, then the maximal number of induced K~(t)'s is achieved in the case of the Tur~n graph T~(n), and that this bound on t is essentially best possible.