On a Problem by Shapozenko on Johnson Graphs - Diego, Víctor; Serra, Oriol; Vena, Lluís (Graphs and combinatorics, 2018)
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 .