Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree
Galvin showed that for all fixed ? and sufficiently large n, the n-vertex graph with minimum degree ? that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n-vertex bipartite graph with minimum degree ? admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree ? admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree ? whose minimum degree drops on deletion of an edge or a vertex.
independent set stable set enumeration