The number of independent sets in a graph with small maximum degree

Abstract

Let $\mathrm{ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$\mathrm{ind}(G)\le 2^{\mathrm{iso}(G)}\prod _{uv\in E(G)}\mathrm{ind}(K_{d(u),d(v)}{)}^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot )$ is vertex degree, $\mathrm{iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ 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 $1\le d\le 5$ then $$\mathrm{ind}(G)\le {\left(2^{d+1}-1\right)}^{\frac{|V(G)|}{2d}},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. 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.

Article will appear in {\em Graphs and Combinatorics}

References in corpus (1)

• The Number of Independent Sets in a Regular Graph

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• Enumerating independent sets in Abelian Cayley graphs