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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) ≤ ∆ for every countable, connected graph G with finite maximum degree ∆ except for three small cycles. We prove that D (G) ≤ √ ∆ + 1 if additionally G does not have pendant edges.doi:10.26493/1855-3974.1852.4f7 fatcat:hock5nu5jrc3ndgqyxijzjpbd4