On a problem by Shapozenko on Johnson graphs
release_7ryai42b4nf2fjatc3i6wfurle
by
Víctor Diego, Oriol Serra, Lluís Vena
2017
Abstract
The Johnson graph J(n,m) has the m--subsets of {1,2,...,n} as
vertices and two subsets are adjacent in the graph if they share m-1
elements. Shapozenko asked about the isoperimetric function μ_n,m(k) of
Johnson graphs, that is, the cardinality of the smallest boundary of sets with
k vertices in J(n,m) for each 1< k<n m. We give an upper
bound for μ_n,m(k) and show that, for each given k such that the
solution to the Shadow Minimization Problem in the Boolean lattice is unique,
and each sufficiently large n, the given upper bound is tight. We also show
that the bound is tight for the small values of k< m+1 and for all values
of k when m=2.
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