Total Weight Choosability of Trees

A total-weighting of a graph $G=(V,E)$ is a mapping $f$ which assigns to each element $y\in V\cup E$ a real number $f(y)$ as the weight of $y$. A total-weighting $f$ of $G$ is proper if the coloring $\phi_{f}$ of the vertices of $G$ defined as $\phi_{f}(v)=f(v)+\sum_{e\in E(v)}f(e)$ is a proper coloring of $G$, i.e., $\phi_{f}(v)\ne\phi_{f}(u)$ for any edge $uv$, where $E(v)$ is the set of edges of $G$ incident to $v$. For positive integers $k$ and $k'$, a graph $G$ is called $(k,k')$-total-weight-choosable if whenever each vertex $v$ is given $k$ permissible weights and each edge $e$ is given $k'$ permissible weights, there is a proper total-weighting $f$ of $G$ which uses only permissible weights on each element $y\in V\cup E$. It is known that every tree is (2,2)-total-weight-choosable and every tree other than $K_2$ is (1,3)-total-weight-choosable. However, the problem of determining which trees are (1,2)-total-weight-choosable remained open. This paper solves this problem and characterizes all (1,2)-...