k5-Free bound for the class of planar graphs

We define k -diverse colouring of a graph to be a proper vertex colouring in which every vertex x sees min { k , d ( x ) } different colours in its neighbors. We show that for given k there is an f ( k ) for which every planar graph admits a k -diverse colouring using at most f ( k ) colours. Then using this colouring we obtain a K 5 -free graph H for which every planar graph admits a homomorphism to it, and thus another proof for the result of J. Nešetřil, P. Ossona de Mendez.