On a heterochromatic number for hypercubes

The neighbourhood heterochromatic numbernh_c(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 lim_n→∞(nh_c(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 2ⁿ-dimensional hypercube.