Locally restricted colorings

We consider the following problem: given suitable integers image and p, what is the smallest value image such that, for any graph G with chromatic number image and any vertex coloring of G with at most image colors, there is a vertex image such that at least image different colors occur within distance image of image? Let image be this value; we show in particular that image for all image. We give the exact value of image when image or image, and image or image.