Combinatorica

, Volume 11, Issue 4, pp 299–314

Edge-isoperimetric inequalities in the grid

  • Béla Bollobás
  • Imre Leader
Article

DOI: 10.1007/BF01275667

Cite this article as:
Bollobás, B. & Leader, I. Combinatorica (1991) 11: 299. doi:10.1007/BF01275667

Abstract

The grid graph is the graph on [k]n={0,...,k−1}n in whichx=(xi)1n is joined toy=(yi)1n if for somei we have |xi−yi|=1 andxj=yj for allji. In this paper we give a lower bound for the number of edges between a subset of [k]n of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA⊂[k]n satisfieskn/4≤|A|≤3kn/4 then there are at leastkn−1 edges betweenA and its complement.

Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.

We also give a best possible upper bound for the number of edges spanned by a subset of [k]n of given cardinality. In particular, forr=1,...,k we show that ifA⊂[k]n satisfies |A|≤rn then the subgraph of [k]n induced byA has average degree at most 2n(1−1/r).

AMS subject classification (1991)

05 C 35

Copyright information

© Akadémiai Kiadó 1991

Authors and Affiliations

  • Béla Bollobás
    • 1
    • 2
  • Imre Leader
    • 1
    • 2
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeEngland
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA