An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index of $G$, denoted by ${\chi }_a^{{}^{\prime }}(G)$, is the smallest integer $k$ such that $G$ is acyclically edge $k$-colorable. In this paper, we consider the planar graphs without 3-cycles and intersecting 4-cycles, and prove that ${\chi }_a^{{}^{\prime }}(G)\le \mathrm{\Delta }(G)+1$ if $\mathrm{\Delta }(G)\ge 8$.

An injective coloring of a graph $G$ is a vertex coloring such that a pair of vertices obtain distinct colors if there is a path of length two between them. It is proved in this paper that ${\chi }_i^l(G)\le \mathrm{\Delta }+4$ if $\mathrm{\Delta }\ge 12$ when $G$ does not have a $4^{-}$-cycle intersecting with a $5^{-}$-cycle. Our result improves a previous result of Cai et al. in 2023, who showed that ${\chi }_i^l(G)\le \mathrm{\Delta }+4$ when $\mathrm{\Delta }\ge 12$ and $G$ has disjoint $5^{-}$-cycles.