Hitting all maximum independent sets

Abstract

We describe an infinite family of graphs G n , where G n has n vertices, independence number at least n/4, and no set of less than √ n/2 vertices intersects all its maximum independent sets. This is motivated by a question of Bollobás, Erdős and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph G on n vertices with independence number (1/4 + ε)n, the average independence number of an induced subgraph of G on a uniform random subset of the vertices is at most (1/4+ε−Ω(ε 2 ))n. Background and resultsThe following conjecture appears in a recent paper of Friedgut, Kalai and Kindler.Conjecture 1.1 ([8], Conjecture 3.1). For every α ∈ (0, 1/2) there exists k and τ > 0 such that if G is a graph on n vertices with maximum independent set of size αn, then there exist pairwise disjoint subsets of vertices A 1 , A 2 , . . . , A r in G such that