An upper bound on the distinguishing index of graphs with minimum degree at least two

The distinguishing index of a simple graph $G$, denoted by $D^{\mathrm{\prime }}(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pil'sniak (2015) that for any 2-connected graph $D^{\mathrm{\prime }}(G)\le \lceil \sqrt{\mathrm{\Delta }(G)}\rceil +1$. We prove a more general result for the distinguishing index of graphs with minimum degree at least two from which the conjecture follows. Also we present graphs $G$ for which $D^{\mathrm{\prime }}(G)\le \lceil \sqrt{\mathrm{\Delta }}\rceil$.