  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 $\delta \geq
\Delta/2$.
