Cubic graphs with small independence ratio

TL;DR

This paper improves the upper bound on the independence ratio of large girth cubic graphs from 0.45537 to 0.454, refining our understanding of the structure of such graphs.

Contribution

The paper provides a tighter upper bound on the independence ratio for 3-regular graphs with large girth, advancing previous bounds established over the past decades.

Findings

Improved upper bound on i(3,∞) to 0.454

Refined understanding of independence ratios in cubic graphs

Builds on decades of bounds with a new tighter estimate

Abstract

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \ge 0.4375$, and Cs\'oka improved it to $i(3,\infty) \ge 0.44533$ in 2016. Bollob\'as proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$ in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \le 0.454.$