General and acyclic sum-list-colouring of graphs

We investigate list colouring of a graph in which the sizes of the lists assigned to different vertices can be different. For a given graph G and a class of graphs P we colour G from the lists in such a way that each colour class induces a graph in P . The aim is to find the P-sum-choice-number of G, which means the smallest possible sum of all the list sizes such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes. We prove several general results concerning the P-sum-choice-number of an arbitrary graph. Using some of them, we also estimate or, in the case of complete graphs or some complete bipartite graphs, exactly determine the P-sum-choice-number of a graph, when P is the class of acyclic graphs.