Acyclic sum-list-colouring of grids and other classes of graphs

In this paper we consider list colouring of a graph G in which the sizes of lists assigned to different vertices can be different. We colour G from the lists in such a way that each colour class induces an acyclic graph. The aim is to find 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. This invariant is called the D1-sum-choice-number of G. In the paper we investigate the D1-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the D1-sum-choice-number for each grid [formula], when at least one of the numbers n, rn is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the D1-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs.