We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/α(G) \leq 4 - \varepsilon$ 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：我们证明了每个不含与4-环共享边的三角形的$n$顶点平面图$G$，其独立率满足$n/α(G) \leq 4 - \varepsilon$，其中$\varepsilon = 1/30$。该结果意味着相同的界适用于4-无环平面图，以及无相邻三角形且不含与5-环共享边的三角形的平面图。对于后一种情况，我们将界加强至$\varepsilon = 2/9$。