Independent Sets of Random Trees and of Sparse Random Graphs

Abstract

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 (G) be the number of independent sets of size k in the graph G and let α(G) = max{k ≥ 0 : x k (G) = 0}. In 1987, Alavi, Malde, Schwenk and Erdös asked if the independent set sequence x 0 (G), x 1 (G), . . . , x α(G) (G) 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 Erdös is "four-fifths true", with high probability.We also show unimodality of the independent set sequence of Erdös-Renyi 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.

On the independent set sequence of a tree

“…Bootstrapping in this way, we eventually get to M = 400000, at which value (16) holds for p ≤ 112384, yielding the bound claimed in Theorem 1.7. All computations were performed on Mathematica.…”

“…With some further computation it is likely that we could improve to ℓ = 0.28098n, but not to 0.28099n. Using quite different methods Heilman [16] has recently shown that the independent set sequence of T is a.a.s. weakly increasing up to 0.26543n, or for the initial about 46% of its non-zero part.…”