  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 $i(3,\infty) \le 0.454.$
