  A proper vertex-colouring of a graph G is said to be locally identifying if
for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the
sets of colours in the closed neighbourhoods of u and v are different. We show
that any graph G has a locally identifying colouring with $2\Delta^2-3\Delta+3$
colours, where $\Delta$ is the maximum degree of G, 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.
