List improper colourings of planar graphs

A graph G is m-choosable with impropriety d, or simply (m,d)*-choosable, if for every list assignment L, where \L(upsilon)\ greater than or equal to m for every upsilon is an element of V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the game colour as itself. We show that every planar graph is (3, 2)*-choosable and every outerplanar graph is (2, 2)*-choosable. We also propose some interesting problems about this colouring.