Independent Sets in Graphs with given Minimum Degree

The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let i(G) be the number of independent sets in a graph G and let it(G) be the number of independent sets in G of size t. Kahn used entropy to show that if G is an r-regular bipartite graph with n vertices, then i(G) ≤ i(Kr,r). Zhao used bipartite double covers to extend this bound to general r-regular graphs. Galvin proved that if G is a graph with δ(G) ≥ δ and n large enough, then i(G) ≤ i(Kδ,n−δ). In this paper, we prove that if G is a bipartite graph on n vertices with δ(G) ≥ δ where n ≥ 2δ, then it(G) ≤ it(Kδ,n−δ) when t ≥ 3. We note that this result cannot be extended to t = 2 (and is trivial for t = 0, 1). Also, we use Kahn’s entropy argument and Zhao’s extension to prove that if G is a graph with n vertices, δ(G) ≥ δ, and ∆(G) ≤ ∆, then i(G) ≤ i(Kδ,∆).