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Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs. (English) Zbl 0906.05066
The work can be considered as continuation of F. R. K. Chung, R. L. Graham and R. M. Wilson [Combinatorica 9, No. 4, 345-362 (1989; Zbl 0715.05057)] on quasi-random properties of graphs. A $$G_n$$ sequence of graphs is quasi-random if it exhibits some behavior of random graph properties asymptotically, and it turns out that a number of equivalent properties ensure this. Informally these assumtions can be summerized as $$G_n$$ contains about the same number of subgraphs as the appropriate $$G_{n, p}$$ does. However, assumptions on certain subgraphs do not imply quasi randomness, such as the subgraphs having 3 vertices, the odd cycles, or the cutsets of size $$\lfloor n/2 \rfloor$$, where $$n$$ is the number of vertices. The main idea of the paper is that the properties above still imply the quasi randomness of $$G_n$$ if one imposes them not only on the members of $$G_n$$ but on the induced subgraphs also. The proofs heavily use the so-called Szemerédi partitions and the celebrated regularity lemma.

##### MSC:
 05C80 Random graphs (graph-theoretic aspects) 05C99 Graph theory
##### Keywords:
quasi-random graphs; graph properties; regularity lemma
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##### References:
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