Counting independent sets of a fixed size in graphs with a given minimum degree

TL;DR

This paper investigates which graphs maximize the number of independent sets of a fixed size given a minimum degree, extending previous results and focusing on specific degree and size ranges.

Contribution

It proves that for certain degree and size parameters, the complete bipartite graph maximizes the count of independent sets, advancing the understanding of extremal graph configurations.

Findings

For $oldsymbol{oldsymbol{ ext{all } oldsymbol{ ext{triples }(n,oldsymbol{ ext{ extdelta}},t)}}}$ with $ ext{ extdelta} oldsymbol{ ext{ extleq}} 3$ and $t oldsymbol{ ext{ extgeq}} 3$, $K_{ ext{ extdelta}, n- ext{ extdelta}}$ maximizes independent sets.

For $oldsymbol{ ext{ extdelta} > 3}$ and $t oldsymbol{ ext{ extgeq}} 2 ext{ extdelta} + 1$, the same extremal graph is optimal.

The study introduces a family of critical graphs where minimum degree drops upon deletion of edges or vertices.

Abstract

Galvin showed that for all fixed $\delta$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $\delta$ that admits the most independent sets is the complete bipartite graph $K_{\delta,n-\delta}$. 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 $(n,\delta, t)$ with $t\geq 3$, no $n$-vertex {\em bipartite} graph with minimum degree $\delta$ admits more independent sets of size $t$ than $K_{\delta,n-\delta}$. Here we make further progress. We show that for all triples $(n,\delta,t)$ with $\delta \leq 3$ and $t\geq 3$, no $n$-vertex graph with minimum degree $\delta$ admits more independent sets of size $t$ than $K_{\delta,n-\delta}$, and we obtain the same conclusion for $\delta > 3$ and $t \geq 2\delta +1$. Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree $\delta$ whose minimum degree drops on deletion of an edge or a vertex.