The Number of Independent Sets in a Graph with Small Maximum Degree

Let ind(G) be the number of independent sets in a graph G. We show that if G has maximum degree at most 5 then ind(G) <= 2(iso(G)) Pi(uv is an element of E(G)) ind(K-d(u),K-d(v))(1/d(u)d(v)) (where d(.) is vertex degree, iso(G) is the number of isolated vertices in G and K-a,K-b is the complete bipartite graph with a vertices in one partition class and b in the other), with equality if and only if each connected component of G is either a complete bipartite graph or a single vertex. This bound (for all G) was conjectured by Kahn. A corollary of our result is that if G is d-regular with 1 <= d <= 5 then ind(G) <= (2(d+1) - 1)(vertical bar V(G)vertical bar/2d), with equality if and only if G is a disjoint union of vertical bar V(G)vertical bar/2d copies of K-d,K-d. This bound ( for all d) was conjectured by Alon and Kahn and recently proved for all d by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most 3 the search could be done by hand, but for the case of maximum degree 4 or 5, a computer is needed.