On a heterochromatic number for hypercubes

The neighbourhood heterochromatic number  nhc(G)nhc(G) of a non-empty graph G is the smallest integer l such that for every colouring of G with exactly l colours, G   contains a vertex all of whose neighbours have different colours. We prove that limn→∞(nhc(Gn)-1)/|V(Gn)|=1limn→∞(nhc(Gn)-1)/|V(Gn)|=1 for any connected graph G   with at least two vertices. We also give upper and lower bounds for the neighbourhood heterochromatic number of the 2n2n-dimensional hypercube.