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The Vertex Isoperimetric Problem on Kneser Graphs

For a simple graph G = (V,E), the vertex boundary of a subset A ⊆ V consists of all vertices not in A that are adjacent to some vertex in A. The goal of the vertex isoperimetric problem is to determine the minimum boundary size of all vertex subsets of a given size. In particular, define μG(r) as the minimum boundary size of all vertex subsets of G of size r. Meanwhile, the vertex set of the Kneser graph KGn,k is the set of all k-element subsets of {1, 2, . . . , n}, and two vertices are adjacent if their corresponding sets are disjoint. The main results of this paper are to compute μG(r) for small and large values of r, and to prove the general lower bound μG(r) ≥ ( n k ) − 1r ( n−1 k−1 )2−r when G = KGn,k. We will also survey the vertex isoperimetric problem on a closely related class of graphs called Johnson graphs and survey cross-intersecting families, which are closely related to the vertex isoperimetric problem on Kneser graphs.