The independence ratio of 4-cycle-free planar graphs

We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n\mathrm{/}\alpha (G)\le 4-\epsilon$ for $\epsilon =1\mathrm{/}30$. This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to $\epsilon =2\mathrm{/}9$.