2015年Cardoso等人在探究电台网络打包(PRN)问题时给出了injective-边染色的概念。图的$k$-injective-边染色是指对于图$G$给定一个边染色$f: E(G)\rightarrow C=\{1,2,\cdots,k\}$, 若$e_{1},\ e_{2},\ e_{3}$是$G$中连续的3条边, 则有$f(e_{1})\neq f(e_{3})$。图$G$的injective-边染色数是指使得图$G$存在一个$k$-injective-边染色的最小整数$k$, 用$\chi'_{i}(G)$表示。本文运用极小反例和权转移方法证明了: 对不包含$k$-圈且$4^{-}$-圈互不相交的平面图$G$, 有$\chi'_{i}(G)\leqslant 3\Delta(G)-2$, 其中$5\leq k\leq10$。

To explore radio network packaging issues, Cardoso et al. proposed the concept of injective edge coloring of a graph in 2015. A $k$-injective edge coloring of $G$ is a coloring of the edges of $G, f$: $E(G)\to C=\{1,2,\cdots,k\}$, such that if $e_{1},\ e_{2},\ e_{3}$ are three consecutive edges in $G$, then $f(e_{1})\neq f(e_{3})$, where $e_{1},\ e_{2}$ and $e_{3}$ are consecutive if they form a path or a cycle of length 3. The injective edge coloring number, denoted by $\chi'_{i}(G)$, is the minimum number of colors such that $G$ has such a coloring. In this paper, we use minimal counter example and power transfer method to prove that if $G$ is a planar graph without $k$-cycles and $4^{-}$-cycles disjoint, then $\chi'_{i}(G)\leq3\Delta(G)-2$, where $5\leq k\leq10$.