List packing of graphs with bounded tree-width

Assume $L$ is a $k$-assignment of a graph $G$. An $L$-packing $\phi$ of $G$ is a sequence $\phi=(\phi_1, \ldots, \phi_k)$ of $k$-mappings such that each $\phi_i$ is an $L$-coloring of $G$, and for each vertex $v$ of $G$, $\{\phi_1(v), \ldots, \phi_k(v)\} = L(v)$ (and hence $\phi_i(v) \ne \phi_j(v)$ when $i \ne j$). We say $G$ is list $k$-packable if for any $k$-assignment $L$ of $G$, there is an $L$-packing of $G$. The list packing number $\chi_l^{\star}(G)$ of $G$ is the minimum integer $k$ such that $G$ is $k$-packable. For a positive integer $d$, let $t(d)$ be the maximum packing number of graphs of tree-width at most $d$. It was known that $d+1 \le t(d) \le 2d$ for any $d$. In this paper, we prove that $t(d) \le 2d-1$ for $d \ge 3$, and $t(d) \ge d+2$ for $d \ge 2$. In particular, $t(2)=4$ and $t(3)=5$. Furthermore, we show that for constant positive integers $k, d$, the problem of determining $\chi_l^{\star}(G)\leq k$ or not for a graph $G$ of tree-width at most $d$ is solvable in linear time.