A class of graphs with distinguishing index ${\mathbf{D}}^{\mathbf{\prime }}\le \mathbf{3}$

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 $\mathrm{\Delta }$ satisfy $\delta \geq \Delta/2$.