Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Galvin showed that for all fixed δ and sufficiently large n , the n ‐vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph K δ , n − δ . 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 ( n , δ , t ) with t ≥ 3 , no n ‐vertex bipartite graph with minimum degree δ admits more independent sets of size t than K δ , n − δ . Here, we make further progress. We show that for all triples ( n , δ , t ) with δ ≤ 3 and t ≥ 3 , no n ‐vertex graph with minimum degree δ admits more independent sets of size t than K δ , n − δ , and we obtain the same conclusion for δ > 3 and t ≥ 2 δ + 1 . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.