Abstract :
It is known that if every vertex v of an outerplanar graph G is given a list L(v) of at least two colours, then G has an (L,2)∗-colouring; that is, one can colour each vertex with a colour from its own list so that no vertex has more than two neighbours with the same colour as itself. It is proved here that if, in addition, |L(u)∩L(v)|⩽1 for each edge uv, or |L(u)∪L(v)|⩾4 for each edge uv, then G has an (L,1)∗-colouring, and if |L(u)∪L(v)|⩾5 for each edge uv then G has an (L,0)∗-colouring (a proper L-colouring). All possible choosability results of these types for outerplanar graphs, K2,3-minor-free graphs and K4-minor-free graphs are also described here.
