### Abstract

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most (2d+1-1)n/2d by the Kahn-Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for n≥2d, the number of independent sets in a graph with δ(G)≥d is at most that in K_{d,n-d}.In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn-Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvin's conjecture, covering the case n≤2d as well. We find it convenient to address this problem from the perspective of G-. From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with δ(G)≤r, where n=a(r+1)+b with 0≤b≤r, is bounded above by the number of complete subgraphs in aK_{r+1}∪K_{b}.

Original language | English |
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Pages (from-to) | 60-71 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 104 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2014 |

### Keywords

- Complete subgraphs
- Extremal enumeration
- Independent sets

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*104*(1), 60-71. https://doi.org/10.1016/j.jctb.2013.10.003