Injective Edge Coloring for Graphs with Small Edge Weight

An injective k-edge coloring of a graph G = (V(G), E(G)) is a k-edge coloring phi such that if e(1) and e(2) are at distance exactly 2 or in the same triangle, then phi(e(1)) not equal phi(e(2)). The injective chromatic index of G, denoted by chi(i)'(G), is the minimum k such that G has an injective k-edge coloring. The edge weight of G is defined as ew(G) = max{d(G)(u)+d(G)(v):uv is an element of E(G)}. In this paper, we show that chi(i)'(G) <= 3 if ew(G) <= 5; chi(i)'(G) <= 7 if ew(G) <= 6; and chi(i)'(G) <= 11 if ew(G) <= 7.