The effect of induced subgraphs on quasi-randomness

Asaf Shapira, Raphael Yuster*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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 (Combinatorica 9 (1989), 345-362) call a graph p-quasi-random if it satisfies a long list of the properties that hold in G(n,p) with high probability, such as edge distribution, spectral gap, cut size, and so on. 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 p-quasi-random, where p is the unique nontrivial solution of the polynomial equation xδ(1 - x)1-δ = pδ(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.

Original languageEnglish
Pages (from-to)90-109
Number of pages20
JournalRandom Structures and Algorithms
Volume36
Issue number1
DOIs
StatePublished - Jan 2010
Externally publishedYes

Keywords

  • Induced subgraph
  • Quasi-randomness
  • Random graph

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