Stars on trees

For a positive integer r and a vertex v of a graph G, let IG(r)(v) denote the set of independent sets of G that have exactly r elements and contain v. Motivated by a problem of Holroyd and Talbot, Hurlbert and Kamat conjectured that for any r and any tree T, there exists a leaf z of T such that |IT(r)(v)|≤|IT(r)(z)| for each vertex v of T. They proved the conjecture for r≤4. We show that for any integer k≥3, there exists a tree Tk that has a vertex x such that x is not a leaf of Tk, |ITk(r)(z)|<|ITk(r)(x)| for any leaf z of Tk and any integer r with 5≤r≤2k+1, and 2k+1 is the largest integer s for which ITk(s)(x) is non-empty.