The independent set sequence of regular bipartite graphs

TL;DR

This paper investigates the unimodality of the independent set sequence in regular bipartite graphs, providing bounds and asymptotic estimates, especially for hypercubes, to support conjectures about unimodality.

Contribution

It offers partial unimodality results for regular bipartite graphs and hypercubes, with new bounds and asymptotic estimates for independent set counts.

Findings

Partial unimodality results for regular bipartite graphs

Asymptotically tight estimates for hypercube independent sets

Stronger unimodality bounds for hypercube sequences

Abstract

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite $G$. We consider the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges). Using bounds on the independent set polynomial $P(G,\lambda):=\sum_{t \geq 0} i_t(G)\lambda^t$ for these graphs, we obtain partial unimodality results in these cases. We then focus on the discrete hypercube $Q_d$, the graph on vertex set $\{0,1\}^d$ with two strings adjacent if they differ on exactly one coordinate. We obtain asymptotically tight estimates for $i_{t(d)}(Q_d)$ in the range $t(d)/2^{d-1} > 1-1/\sqrt{2}$, and nearly matching upper and lower bounds otherwise. We use these estimates to obtain a stronger partial unimodality result for the independent set sequence of $Q_d$.