The fractional list packing number $χ_{\ell}^{\bullet}(G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ need to be to ensure the existence of a `perfectly balanced' probability distribution on proper $L$-colourings, i.e., such that at every vertex $v$, every colour appears with equal probability $1/|L(v)|$. In this work we give various bounds on $χ_{\ell}^{\bullet}(G)$, which admit strengthenings for correspondence and local-degree versions. As a corollary, we improve theorems on the related notion of flexible list colouring. In particular we study Cartesian products and $d$-degenerate graphs, and we prove that $χ_{\ell}^{\bullet}(G)$ is bounded from above by the pathwidth of $G$ plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.

翻译：图$G$的分式列表打包数$χ_{\ell}^{\bullet}(G)$是近期源于不相交列表染色研究的一个图不变量。它衡量列表赋值$L:V(G)\rightarrow 2^{\mathbb{N}}$需要多大才能确保存在一个关于正常$L$-染色的"完全平衡"概率分布，即在每个顶点$v$处，每种颜色以相等概率$1/|L(v)|$出现。本文给出了$χ_{\ell}^{\bullet}(G)$的多种界，这些界可通过对应版本和局部度版本得到加强。作为推论，我们改进了关于柔性列表染色这一相关概念的定理。特别地，我们研究了笛卡尔积图和$d$-退化图，并证明了$χ_{\ell}^{\bullet}(G)$的上界为$G$的路径宽度加一。该结论的对应版本在树宽（而非路径宽）情形下不成立。