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A proper vertex-colouring of a graph G is said to be locally identifying iffor any pair u,v of adjacent vertices with distinct closed neighbourhoods, thesets of colours in the closed neighbourhoods of u and v are different. We showthat 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 waya question asked by Esperet et al. We also provide similar results for locallyidentifying colourings which have the property that the colours in theneighbourhood of each vertex are all different and apply our method to theclass of chordal graphs.

Locally identifying colourings for graphs with given maximum degree