Contract grant sponsor: National Security Agency; contract grant numbers: H98230-10-1-0364 and H98230-13-1-0248; contract grant sponsor: Simons Foundation.
Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree
Article first published online: 21 AUG 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 76, Issue 2, pages 149–168, June 2014
How to Cite
Engbers, J. and Galvin, D. (2014), Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree. J. Graph Theory, 76: 149–168. doi: 10.1002/jgt.21756
- Issue published online: 1 APR 2014
- Article first published online: 21 AUG 2013
- Manuscript Revised: 2 JUL 2013
- Manuscript Received: 24 APR 2012
- National Security Agency. Grant Numbers: H98230-10-1-0364, H98230-13-1-0248
- Simons Foundation
- independent set;
- stable set;
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 . 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 δ 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 δ 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 δ whose minimum degree drops on deletion of an edge or a vertex.