Abstract
The grid graph is the graph on [k]n={0,...,k−1}n in whichx=(x i ) n1 is joined toy=(y i ) n1 if for somei we have |x i −y i |=1 andx j =y j for allj≠i. 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 satisfiesk n/4≤|A|≤3k n/4 then there are at leastk n−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|≤r n then the subgraph of [k]n induced byA has average degree at most 2n(1−1/r).
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References
Bernstein, A. J.: Maximally connected arrays on then-cube,SIAM J. Appl. Math. 15 (1967), 1485–1489.
Bollobás, B.:Combinatorics, Cambridge University Press, 1986, xii+177 pp.
Bollobás, B., andLeader, I.: Compressions and isoperimetric inequalities,J. Combinatorial Theory (A),56 (1991), 47–62.
Clements, G. F.: Sets of lattice points which contain a maximal number of edges,Proc. Amer. Math. Soc. 27 (1971), 13–15.
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.
Harper, L. H.: Optimal assignments of numbers to vertices,SIAM J. Appl. Math. 12 (1964), 131–135.
Hart, S. A note on the edges of then-cube,Discrete Math. 14 (1976), 157–163.
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.
Lindsey, J. H.: Assignment of numbers to vertices,Amer. Math. Monthly 71 (1964), 508–516.
Wang, D.-L., andWang, P.: Discrete isoperimetric problems,SIAM J. Appl. Math 32 (1977), 860–870.
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Research partially supported by NSF Grant DMS-8806097
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Bollobás, B., Leader, I. Edge-isoperimetric inequalities in the grid. Combinatorica 11, 299–314 (1991). https://doi.org/10.1007/BF01275667
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DOI: https://doi.org/10.1007/BF01275667