Quasi-randomness of graph balanced cut properties

Hao Huang, Choongbum Lee

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of graphs. Let k ≥ 2 be a fixed integer, α 1,...,α k be positive reals satisfying ∑ iα i = 1 and (α 1,...,α k)≠(1/k,...,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V 1,...,V k of size α 1n,...,α kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi-random. However, the method of quasi-random hypergraphs they used did not provide enough information to resolve the case (1/k,...,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi-random. Janson also posed the same question in his study of quasi-randomness under the framework of graph limits. In this paper, we positively answer their question.

Original languageEnglish (US)
Pages (from-to)124-145
Number of pages22
JournalRandom Structures and Algorithms
Volume41
Issue number1
DOIs
StatePublished - Aug 1 2012

Keywords

  • Cut properties
  • Pseudo-random graphs
  • Quasi-random graphs

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