PROOF OF A CONJECTURE ON THE DISTINGUISHING INDEX OF A GRAPH

The distinguishing index of a graph G, denoted by D 0 (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 0 (G) ≤ d p Δ(G)e + 1. We prove this conjecture. Furthermore, we propose a more general conjecture for upper bound of D 0 (G).