On the density of a graph and its blowup

Asaf Shapira, Raphael Yuster

Research output: Contribution to journalArticlepeer-review

Abstract

It is well known that, of all graphs with edge-density p, the random graph G(n,p) contains the smallest density of copies of Kt,t, the complete bipartite graph of size 2 t. Since Kt,t is a t-blowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle-density p3, the random graph G(n,p) contains close to the smallest density of Kt,t,t, which is the t-blowup of a triangle? Our main result gives an indication that the answer to the above question is positive by showing that for some blowup, the answer must be positive. More formally we prove that if G has triangle-density p3, then there is some 2≤t≤T(p) for which the density of Kt,t,t in G is at least p(3+o(1))t2, which (up to the o(1) term) equals the density of Kt,t,t in G(n,p). We also raise several open problems related to these problems and discuss some applications to other areas.

Original languageEnglish
Pages (from-to)704-719
Number of pages16
JournalJournal of Combinatorial Theory. Series B
Volume100
Issue number6
DOIs
StatePublished - Nov 2010

Bibliographical note

Funding Information:
E-mail addresses: asafico@math.gatech.edu (A. Shapira), raphy@math.haifa.ac.il (R. Yuster). 1 Research supported by NSF Grant DMS-0901355.

Keywords

  • Blowup
  • Graph density
  • Triangle density

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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