TY - JOUR

T1 - Quasi-randomness and the distribution of copies of a fixed graph

AU - Shapira, Asaf

PY - 2008/11

Y1 - 2008/11

N2 - We show that if a graph G has the property that all subsets of vertices of size n/4 contain the "correct" number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles. A similar improvement of [10] is also obtained for any fixed graph other than the triangle, and for any edge density other than 1/2. The proof relies on a theorem of Gottlieb [7] in algebraic combinatorics, concerning the rank of set inclusion matrices.

AB - We show that if a graph G has the property that all subsets of vertices of size n/4 contain the "correct" number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles. A similar improvement of [10] is also obtained for any fixed graph other than the triangle, and for any edge density other than 1/2. The proof relies on a theorem of Gottlieb [7] in algebraic combinatorics, concerning the rank of set inclusion matrices.

UR - http://www.scopus.com/inward/record.url?scp=65749104074&partnerID=8YFLogxK

U2 - 10.1007/s00493-008-2375-0

DO - 10.1007/s00493-008-2375-0

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AN - SCOPUS:65749104074

SN - 0209-9683

VL - 28

SP - 735

EP - 745

JO - Combinatorica

JF - Combinatorica

IS - 6

ER -