The Distinguishing Index Of Connected Graphs Without Pendant Edges

We consider edge colourings, not necessarily proper. The distinguishing index D' (G) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D' (G) <= Delta for every countable, connected graph G with finite maximum degree A except for three small cycles. We prove that D' G) <= inverted right perpendicular root Delta inverted left perpendicular + 1 if additionally G does not have pendant edges.