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

TL;DR

This paper proves a conjectured upper bound on the number of independent sets in graphs with maximum degree at most 5, characterizing extremal graphs and extending previous conjectures by Kahn and Alon-Kahn.

Contribution

The paper establishes a new upper bound on independent sets for graphs with degree up to 5, confirming conjectures and characterizing extremal structures.

Findings

Proved the bound for graphs with maximum degree 5.

Characterized extremal graphs achieving equality.

Extended previous conjectures by Kahn and Alon-Kahn.

Abstract

Let ${\rm ind}(G)$ 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 $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ 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 $1 \leq d \leq 5$ 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 $K_{d,d}$. 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.