An injective edge-coloring of a graph G is an edge-coloring \(\phi \) of G, not necessarily proper, such that \(\phi (e_1) \ne \phi (e_2)\) whenever there exists an edge \(e_3\) adjacent to both \(e_1\) and \(e_2\) . In particular, edges in a triangle must receive distinct colors. The minimum number of colors needed in an injective edge-coloring of G, denoted by \(\chi _{inj}^{\prime }(G)\) , is called the injective chromatic index of G. In this paper, we study \(K_4\) -minor free graphs, or equivalently, graphs with treewidth at most 2. Axenovich et al. [1] studied the injective chromatic index of graphs with treewidth at most k, and their result implies that every \(K_4\) -minor free graph has injective chromatic index at most 9. This improves a linear bound established by Lv et al. [6] to a constant. For subcubic graphs, Kostochka et al. [5] showed that every subcubic planar graph has injective chromatic index at most 6, which implies that every subcubic \(K_4\) -minor free graph has injective chromatic index at most 6. In the paper, we study \(K_4\) -minor free graphs with maximum degree at most 4 and prove that every such graph has injective chromatic index at most 7.
