A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if, for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the same colour as itself. We prove a Grötzsch-type theorem for list colourings with impropriety one, that is, the (3, 1)*-choosability for triangle-free planar graphs; in the proof the method of extending a precolouring of a 4- or 5-cycle is used.

A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the same 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.

We study the concept of list total colourings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total choice number of a graph of maximum degree 2 is equal to its total chromatic number.