Graphs and Combinatorics

, Volume 27, Issue 2, pp 177–186

The Number of Independent Sets in a Graph with Small Maximum Degree

Original Paper

Abstract

Let ind(G) be the number of independent sets in a graph G. We show that if G has maximum degree at most 5 then
$${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$
(where d(·) is vertex degree, iso(G) is the number of isolated vertices in G and Ka,b is the complete bipartite graph with a vertices in one partition class and b in the other), with equality if and only if each connected component of G is either a complete bipartite graph or a single vertex. This bound (for all G) was conjectured by Kahn. A corollary of our result is that if G is d-regular with 1 ≤ d ≤ 5 then
$${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$
with equality if and only if G is a disjoint union of |V(G)|/2d copies of Kd,d. This bound (for all d) was conjectured by Alon and Kahn and recently proved for all d by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most 3 the search could be done by hand, but for the case of maximum degree 4 or 5, a computer is needed.

Keywords

Independent set Stable set Regular graph 

Mathematics Subject Classification (2000)

Primary 05C69 Secondary 05A16 

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References

  1. 1.
    Alon N.: Independent sets in regular graphs and sum-free subsets of finite groups. Israel J. Math. 73, 247–256 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Kahn J.: An entropy approach to the hard-core model on bipartite graphs. Combin. Probab. Comput. 10, 219–237 (2001)MATHMathSciNetGoogle Scholar
  3. 3.
    Zhao Y.: The number of independent sets in a regular graph. Combin. Probab. Comput. 19, 315–320 (2010)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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