On a problem by Shapozenko on Johnson graphs

TL;DR

This paper investigates the isoperimetric problem in Johnson graphs, providing bounds and tightness results for the minimal boundary size of vertex subsets, with specific results for small and particular parameter values.

Contribution

It introduces an upper bound for the isoperimetric function of Johnson graphs and proves its tightness in certain cases, connecting to the Shadow Minimization Problem.

Findings

Upper bound for the isoperimetric function   

Bound is tight for small k   and for all k when m=2

Bound is tight for large n when the Shadow Minimization Problem solution is unique

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 $1\le k\le {n\choose m}$. 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 $m=2$.