On Injective Edge-Colorings of Graphs Without a K₄- or a K₅-minor

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) ,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 χ 'ᵢₙⱼ(G) χ inj ′ ( G ) . In this paper, we prove the following: If G is subcubic and K₄ K 4 -minor free, then χ ᵢₙⱼ (G) ≤ 5 χ inj ′ ( G ) ≤ 5 ; and if G is K₅ K 5 -minor free, then χ ᵢₙⱼ (G) ≤ 30 χ inj ′ ( G ) ≤ 30 .