Injective edge-coloring of sparse graphs

An injective edge-coloring $c$ of a graph $G$ is an edge-coloring such that if $e_1$, $e_2$, and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive different colors. The minimum integer $k$ such that, $G$ has an injective edge-coloring with $k$ colors, is called the injective chromatic index of $G$ ($\chi'_{\textrm{inj}}(G)$). This parameter was introduced by Cardoso et \textit{al.} \cite{CCCD} motivated by the Packet Radio Network problem. They proved that computing $\chi'_{\textrm{inj}}(G)$ of a graph $G$ is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. The obtained general bound gives 8 for the injective chromatic index of a subcubic graph. If the graph is subcubic bipartite we improve this last bound. We prove that a subcubic bipartite graph has an injective chromatic index bounded by $6$. We also prove that if $G$ is a subcubic graph with maximum average degree less than $\frac{7}{3}$ (resp. $\frac{8}{3}$, $3$), then $G$ admits an injective edge-coloring with at most 4 (resp. $6$, $7$) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs

Injective edge-coloring of graphs with given maximum degree

A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $\chi'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index.Comment: 13 page