Exponential Independence in Subcubic Graphs

TL;DR

This paper introduces the concept of exponential independence in graphs, establishes bounds for subcubic graphs, and explores properties of exponential independent sets in various graph classes.

Contribution

It defines exponential independence, provides bounds on exponential independence numbers in subcubic graphs, and investigates related open problems.

Findings

Subcubic graphs of order n have exponential independent sets of size Ω(n/ log^2 n)

The infinite cubic tree has no exponentially independent set of positive density

Subcubic trees of order n have exponentially independent sets of size (n+3)/4

Abstract

A set $S$ of vertices of a graph $G$ is exponentially independent if, for every vertex $u$ in $S$, $$\sum\limits_{v\in S\setminus \{ u\}}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}<1,$$ where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u$ and $v$ in the graph $G-(S\setminus \{ u,v\})$. The exponential independence number $\alpha_e(G)$ of $G$ is the maximum order of an exponentially independent set in $G$. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order $n$ have exponentially independent sets of order $\Omega(n/\log^2(n))$, that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order $n$ have exponentially independent sets of order $(n+3)/4$.