Some totally 4-choosable multigraphs

It is proved that if G is multigraph with maximum degree 3, and every submultigraph ofG has average degree at most 2 1 2 and is different from one forbidden configuration C 4 with average degree exactly 2 1 2 , then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that no two adjacent or incident elements are coloured with the same colour. This shows that the List-Total-Colouring Conjecture, that ch′′(G) = χ(G) for every multigraph G, is true for all multigraphs of this type. As a consequence, if G is a graph with maximum degree 3 and girth at least 10 that can be embedded in the plane, projective plane, torus or Klein bottle, then ch′′(G) = χ(G) = 4. Some further total choosability results are discussed for planar graphs with sufficiently large maximum degree and girth.