A class of graphs with distinguishing index $\bf D' \leq 3$

An edge-coloring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Lehner, Pil\'{s}niak, and Stawiski proved that all connected regular graphs except $K_2$ admit an asymmetric edge-coloring with three colors. We generalize this result for graphs whose minimal degree $\delta$ and the maximal degree $\Delta$ satisfy $\delta \geq \Delta/2$.