  Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of
real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting
function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the
vertex colouring obtained by $c_w(v) = \sum_{e \ni v}w(e)$. We desire the
smallest possible $k$ such that, for any choice of $\{L_e \,|\, e \in E(G)\}$
where $|L_e| \geq k$ for all $e \in E(G)$, there exists an edge weighting
function $w$ for which $c_w$ is proper. The smallest such value of $k$ is the
weight choosability number of $G$.
  This colouring problem, introduced by Bartnicki, Grytczuk and Niwczyk (2009),
is the list variation of the now famous 1-2-3 Conjecture due to Karo\'nski,
{\L}uczak, and Thomason (2004). Bartnicki et al. develop a method for
approaching the problem based on the Combinatorial Nullstellensatz. Though they
show that some particular classes of graphs have weight choosability number at
most $3$, it was known whether their method could be extended to prove a bound
which holds for all admissible graphs. In this paper, we show that this is
indeed possible, showing that every graph is $(\Delta + d + 1)$-weight
choosable, where $\Delta$ is the graph's maximum degree and $d$ is its
degeneracy. In fact, more general results on total weight choosability are
provided, where one assigns weights to edges and vertices. Improved bounds are
also established for some classes of graph products.
