Distinguishing symmetric digraphs by proper arc-colourings of type I

A symmetric digraph G ↔ is obtained from a simple graph G by replacing each edge uv with a pair of opposite arcs u v → , v u → . An arc-colouring c of a digraph G ↔ is distinguishing if the only automorphism of G ↔ preserving the colouring c is the identity. Behzad introduced the proper arc-colouring of type I as an arc-colouring such that any two consecutive arcs u v → , v w → have distinct colours. We establish an optimal upper bound ⌈ 2 Δ ( G ) ⌉ for the least number of colours in a distinguishing proper colouring of type I of a connected symmetric digraph G ↔ . Furthermore, we prove that the same upper bound ⌈ 2 Δ ( G ) ⌉ is optimal for another type of proper colouring of G ↔ , when only monochromatic 2-paths are forbidden.