The super-connectivity of Johnson graphs

Abstract

For positive integers n, k and t, the uniform subset graph G(n, k, t) has all k-subsets of {1, 2, . . . , n} as vertices and two k-subsets are joined by an edge if they intersect at exactly t elements. The Johnson graph J(n, k) corresponds to G(n, k, k − 1), that is, two vertices of J(n, k) are adjacent if the intersection of the corresponding k-subsets has size k − 1. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the superconnectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs J(n, k) for n ≥ k ≥ 1.