  A graph $G$ is total weight $(k,k')$-choosable if for any total list
assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real
numbers, and each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper
total $L$-weighting, i.e., a mapping $f: V(G) \cup E(G) \to \mathbb{R}$ such
that for each $z \in V(G) \cup E(G)$, $f(z) \in L(z)$, and for each edge $uv$
of $G$, $\sum_{e \in E(u)}f(e)+f(u) \ne \sum_{e \in E(v)}f(e) + f(v)$. This
paper proves that if $G$ decomposes into complete graphs of odd order, then $G$
is total weight $(1,3)$-choosable. As a consequence, every Eulerian graph $G$
of large order and with minimum degree at least $0.91|V(G)|$ is total weight
$(1,3)$-choosable. We also prove that any graph $G$ with minimum degree at
least $0.999|V(G)|$ is total weight $(1,4)$-choosable.
