Independent Sets of Random Trees and Sparse Random Graphs

An independent set of size $k$ in a finite undirected graph $G$ is a set of $k$ vertices of the graph, no two of which are connected by an edge. Let $x_k\left(G\right)$ be the number of independent sets of size $k$ in the graph $G$ and let $\alpha \left(G\right)=\max \left\{k\ge 0:x_k\left(G\right)\ne 0\right\}$. In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence $x_0\left(G\right),x_1\left(G\right),\dots ,x_{\alpha \left(G\right)}\left(G\right)$ of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four-fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science.