On Injective Edge-Colorings of Graphs Without a $$K_4$$- or a $$K_5$$-minor

Abstract

Abstract An injective edge-coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges e and f receive distinct colors whenever 1) e and f are in a common triangle; or 2) $$\{e,f\}$$ { e , f } forms a matching and there is a third edge adjacent to both e and f . The minimum number of colors needed for an injective edge-coloring of a graph G is called the injective chromatic index of G , denoted by $$\chi '_{inj}(G)$$ χ inj ′ ( G ) . In this paper, we prove the following: If G is subcubic and $$K_4$$ K 4 -minor free, then $$\chi ^{\prime } _{inj} (G) \le 5$$ χ inj ′ ( G ) ≤ 5 ; and if G is $$K_5$$ K 5 -minor free, then $$\chi ^{\prime } _{inj} (G) \le 30$$ χ inj ′ ( G ) ≤ 30 .