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Keywords:

  • independent set;
  • stable set;
  • enumeration

Abstract

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 inline image. 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 inline image with inline image, no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than inline image. Here, we make further progress. We show that for all triples inline image with inline image and inline image, no n-vertex graph with minimum degree δ admits more independent sets of size t than inline image, and we obtain the same conclusion for inline image and inline image. 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.