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Quasi-randomness and the distribution of copies of a fixed graph. (English) Zbl 1196.05090
The notion of quasi-randomness is due to A. Thomason [Surveys in combinatorics 1987, Pap. 11th Br. Combin. Conf., London/Engl. 1987, Lond. Math. Soc. Lect. Note Ser. 123, 173–195 (1987; Zbl 0672.05068)]. By the subsequent work of F. R. K. Chung, R. L. Graham and R. M. Wilson [Combinatorica 9, No. 4, 345–362 (1989; Zbl 0715.05057)] it turned out that a sequence of graphs is quasi-random if it exhibits some behavior of random graph properties asymptotically, and in fact a number of equivalent properties ensure this. For graphs $$H$$ and $$G$$ and $$U \subset V(G)$$, let $$H(U)$$ be the number of labeled copies of $$H$$ spanned by $$U$$. Two properties that Chung et al gave were $${\mathcal P}_1(t)$$ and $${\mathcal P}_2$$.
M. Simonovits and V. T. Sós [Combinatorica 17, No. 4, 577–596 (1997; Zbl 0906.05066)] gave a different notion, which allowed a generalization of these results. For a fixed graph $$H$$ on $$h$$ vertices, the sequence $$G_n$$ has the property $${\mathcal P}_H$$ if (1) $$H(U)=p^{e(H)}| U| ^h+o(n^h)$$ for all $$U \subset V(G_n)$$. They showed that for any fixed $$H$$ with $$e(H) >0$$, property $$\mathcal P_H$$ is quasi-random. However, it was left open if one really needs the property $${\mathcal P}_H$$ for all $$U \subset V(G_n)$$, and they asked if less assumptions on $$U$$ was enough for quasi-randomness. The author of this paper shows that the quasi-randomness of a sequence $$G_n$$ follows if for a fixed, non-empty graph $$H$$ (1) holds for $$U \subset V(G_n)$$, $$| U| =\lfloor n/(h+1) \rfloor$$, where $$h=| V(H)|$$. The proof utilizes simple counting arguments and an old result of D. H. Gottlieb [Proc. Am. Math. Soc. 17, 1233–1237 (1966; Zbl 0146.01302)] on the rank of incidence matrices.

##### MSC:
 05C80 Random graphs (graph-theoretic aspects) 05C35 Extremal problems in graph theory
##### Keywords:
quasi-random; subgraphs
Full Text:
##### References:
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