TY - GEN

T1 - The effect of induced subgraphs on quasi-randomness

AU - Shapira, Asaf

AU - Yuster, Raphael

PY - 2008

Y1 - 2008

N2 - One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson[9] call a graph p-quasi-random if it satisfies a long list of the properties that hold in G(n, p) with high probability, like edge distribution, spectral gap, cut size, and more. Our main result here is that the following holds for any fixed graph H: if the distribution of induced copies of H in a graph G is close (in a well defined way) to the distribution we would expect to have in G(n, p), then G is either p-quasi-random or p̄-quasirandom, where p̄ is the unique non-trivial solution of the polynomial equation x δ(1 - x) 1-δ = p δ(1 - p) 1-δ, with δ being the edge density of H. We thus infer that having the correct distribution of induced copies of any single graph H, is enough to guarantee that a graph has the properties of a random one. The proof techniques we develop here, which combine probabilistic, algebraic and combinatorial tools, may be of independent interest to the study of quasi-random structures.

AB - One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson[9] call a graph p-quasi-random if it satisfies a long list of the properties that hold in G(n, p) with high probability, like edge distribution, spectral gap, cut size, and more. Our main result here is that the following holds for any fixed graph H: if the distribution of induced copies of H in a graph G is close (in a well defined way) to the distribution we would expect to have in G(n, p), then G is either p-quasi-random or p̄-quasirandom, where p̄ is the unique non-trivial solution of the polynomial equation x δ(1 - x) 1-δ = p δ(1 - p) 1-δ, with δ being the edge density of H. We thus infer that having the correct distribution of induced copies of any single graph H, is enough to guarantee that a graph has the properties of a random one. The proof techniques we develop here, which combine probabilistic, algebraic and combinatorial tools, may be of independent interest to the study of quasi-random structures.

UR - http://www.scopus.com/inward/record.url?scp=58449105123&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:58449105123

SN - 9780898716474

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 789

EP - 798

BT - Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms

T2 - 19th Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 20 January 2008 through 22 January 2008

ER -