A graph $G=(V,E)$ is total weight $(k,k)$-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 proper $L$-total weighting, i.e., a map $phi: V cup E to mathbb{R}$ such that $phi(z) in L(z)$ for $z in V cup E$, and $sum_{e in E(u)}phi(e)+phi(u) e sum_{e in E(v)}phi(e)+phi(v)$ for every edge ${u,v}$. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in [T. Wong and X. Zhu, Total weigt choosability of graphs, J. Graph Th. 66 (2011),198-212] that every nice graph is total weight $(1,3)$-choosable. The problem whether there is a constant $k$ such that every nice graph is total weight $(1,k)$-choosable remained open for a decade and was recently solved by Cao [L. Cao, Total weight choosability of graphs: Towards the 1-2-3 conjecture, J. Combin. Th. B, 149(2021), 109-146], who proved that every nice graph is total weight $(1, 17)$-choosable. This paper improves this result and proves that every nice graph is total weight $(1, 5)$-choosable.
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