On weight choosabilities of graphs with bounded maximum average degree

The well known 1-2-3-Conjecture asserts that every connected graph G of order at least three can be edge-coloured with integers 1 , 2 , 3 so that the sums of colours met by adjacent vertices are distinct in G . The same is believed to hold if instead of edge colourings we consider total colourings assigning 1 or 2 to every vertex and edge of a given graph-this time the colour of every vertex is counted in its corresponding sum. We consider list extensions of the both concepts, where every edge (and vertex) is assigned a set of k positive integers, i.e., its potential colours, and regardless of this list assignment we wish to be able to construct a colouring from these lists so that the adjacent vertices are distinguished by their corresponding sums. We prove that if the maximum average degree of the graph G is smaller than 5 2 , then lists of length k = 3 are sufficient for that goal in case of edge colourings (if G has no isolated edges), while already k = 2 suffices in the total case. In fact the second of these statements remains true with arbitrary real colours admitted instead of positive integers, and the first one-for positive reals. The proofs of these facts are based on the discharging method combined with the algebraic approach of Alon known as Combinatorial Nullstellensatz.