  A graph $G$ is $(k,k')$-choosable if the following holds: For any list
assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real
numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there
is a total weighting
  $\phi: V(G) \cup E(G) \to R$ such that $\phi(z) \in L(z)$ for $z \in V \cup
E$, and $\sum_{e \in E(u)}\phi(e)+\phi(u) \ne \sum_{e \in E(v)}\phi(e)+\phi(v)$
for every edge $uv$. This paper proves the following results: (1) If $G$ is a
connected $d$-degenerate graph, and $k>d$ is a prime number, and $G$ is either
non-bipartite or has two non-adjacent vertices $u,v$ with $d(u)+d(v) < k$, then
$G$ is $(1,k)$-choosable. As a consequence, every planar graph with no isolated
edges is $(1,7)$-choosable, and every connected $2$-degenerate non-bipartite
graph other than $K_2$ is $(1,3)$-choosable. (2) If $d+1$ is a prime number,
$v_1, v_2, \ldots, v_n$ is an ordering of the vertices of $G$ such that each
vertex $v_i$ has back degree $d^-(v_i) \le d$, then there is a graph $G'$
obtained from $G$ by adding at most $d-d^-(v_i)$ leaf neighbours to $v_i$ (for
each $i$) and $G'$ is $(1,2)$-choosable. (3) If $G$ is $d$-degenerate and $d+1$
a prime, then $G$ is $(d,2)$-choosable. In particular, $2$-degenerate graphs
are $(2,2)$-choosable. (4) Every graph is $(\lceil\frac{{\rm
mad}(G)}{2}\rceil+1, 2)$ -choosable. In particular, planar graphs are
$(4,2)$-choosable, planar bipartite graphs are $(3,2)$-choosable.
