On a Problem by Shapozenko on Johnson Graphs

Abstract

The Johnson graph J(n, m) has the m-subsets of $$\{1,2,\ldots ,n\}$$ as vertices and two subsets are adjacent in the graph if they share $$m-1$$ elements. Shapozenko asked about the isoperimetric function $$\mu _{n,m}(k)$$ of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n, m) for each $$1\le k\le {n\atopwithdelims ()m}$$ . We give an upper bound for $$\mu _{n,m}(k)$$ 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 $$k\le m+1$$ and for all values of k when $$m=2$$ .