On a Problem by Shapozenko on Johnson Graphs

Víctor Diego, Oriol Serra, Lluís Vena
2018 Graphs and Combinatorics  
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 µn,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 µn,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
more » ... 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. Johnson graph and Isoperimetric problem and Shift compression.
doi:10.1007/s00373-018-1923-7 fatcat:o762ushckjfmlcvkgbsnnrcmge