K5K5-Free bound for the class of planar graphs

We define kk-diverse colouring of a graph to be a proper vertex colouring in which every vertex xx sees min{k,d(x)}min{k,d(x)} different colours in its neighbors. We show that for given kk there is an f(k)f(k) for which every planar graph admits a kk-diverse colouring using at most f(k)f(k) colours. Then using this colouring we obtain a K5K5-free graph HH for which every planar graph admits a homomorphism to it, and thus another proof for the result of J. Nešetřil, P. Ossona de Mendez.