Cubic Graphs with Small Independence Ratio

  • József Balogh
  • Alexandr Kostochka
  • Xujun Liu


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 $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás 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 $i(3,\infty) \leqslant 0.454.$

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