Combinatorica

, Volume 11, Issue 4, pp 299–314 | Cite as

Edge-isoperimetric inequalities in the grid

  • Béla Bollobás
  • Imre Leader
Article

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bernstein, A. J.: Maximally connected arrays on then-cube,SIAM J. Appl. Math. 15 (1967), 1485–1489.Google Scholar
  2. [2]
    Bollobás, B.:Combinatorics, Cambridge University Press, 1986, xii+177 pp.Google Scholar
  3. [3]
    Bollobás, B., andLeader, I.: Compressions and isoperimetric inequalities,J. Combinatorial Theory (A),56 (1991), 47–62.Google Scholar
  4. [4]
    Clements, G. F.: Sets of lattice points which contain a maximal number of edges,Proc. Amer. Math. Soc. 27 (1971), 13–15.Google Scholar
  5. [5]
    Frankl, P. The shifting technique in extremal set theory, inSurveys in Combinatorics 1987 (Whitehead, C., ed.), London Math. Soc. Lecture Note Series 123, Cambridge University Press, 1987, pp. 81–110.Google Scholar
  6. [6]
    Harper, L. H.: Optimal assignments of numbers to vertices,SIAM J. Appl. Math. 12 (1964), 131–135.Google Scholar
  7. [7]
    Hart, S. A note on the edges of then-cube,Discrete Math. 14 (1976), 157–163.Google Scholar
  8. [8]
    Kleitman, D. J., Krieger, M. M., andRothschild, B. L.: Configurations maximizing the number of pairs of Hamming-adjacent lattice points.Studies in Appl. Math. 50 (1971), 115–119.Google Scholar
  9. [9]
    Lindsey, J. H.: Assignment of numbers to vertices,Amer. Math. Monthly 71 (1964), 508–516.Google Scholar
  10. [10]
    Wang, D.-L., andWang, P.: Discrete isoperimetric problems,SIAM J. Appl. Math 32 (1977), 860–870.Google Scholar

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

Personalised recommendations