Injective Edge-Colorings of $$K_4$$-Minor Free Graphs

Abstract An injective edge-coloring of a graph G is an edge-coloring $$\phi $$ ϕ of G , not necessarily proper, such that $$\phi (e_1) \ne \phi (e_2)$$ ϕ ( e 1 ) ≠ ϕ ( e 2 ) whenever there exists an edge $$e_3$$ e 3 adjacent to both $$e_1$$ e 1 and $$e_2$$ 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 $$\chi _{inj}^{\prime }(G)$$ χ inj ′ ( G ) , is called the injective chromatic index of G . In this paper, we study $$K_4$$ 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 $$K_4$$ 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 $$K_4$$ K 4 -minor free graph has injective chromatic index at most 6. In the paper, we study $$K_4$$ K 4 -minor free graphs with maximum degree at most 4 and prove that every such graph has injective chromatic index at most 7.