Abstract:

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 $5\leq k\leq10$.

Key words: injective edge coloring, planar graph, maximum degree, cycle