Locally identifying colourings for graphs with given maximum degree

A proper vertex-colouring of a graph GG is said to be locally identifying if for any pair uu, vv of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of uu and vv are different. We show that any graph GG has a locally identifying colouring with 2Δ2−3Δ+32Δ2−3Δ+3 colours, where ΔΔ is the maximum degree of GG, answering in a positive way a question asked by Esperet et al. We also provide similar results for locally identifying colourings which have the property that the colours in the neighbourhood of each vertex are all different and apply our method to the class of chordal graphs.