Cubic graphs with small independence ratio

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