We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/\alpha(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. 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 $\varepsilon = 2/9$.

翻译：我们证明每个不含与4-圈共边三角形的$n$顶点平面图$G$满足独立比$n/\alpha(G) \leq 4 - \varepsilon$，其中$\varepsilon = 1/30$。这一结果表明，相同的界对不含4-圈的平面图以及不含相邻三角形且无三角形与5-圈共边的平面图均成立。对于后一类图，我们将界加强至$\varepsilon = 2/9$。