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

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 $t\ge 3$, 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 $\delta \le 3$ and $t\ge 3$, 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 $t\ge 2\delta +1$. 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.

21 pages, 6 figures