A problem of Shapozenko on Johnson graphs

The Johnson graph J(n,m) has the m-subsets of [n] as vertices and two subsets are adjacent in the graph if they share m−1 elements. Shapozenko asked about the isoperimetric function of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n,m) for each . We give an upper bound for the isoperimetric function of Johnson graphs. We show that, for given t and all sufficiently large n, the given bound is tight for . 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.