Fractional list packing for layered graphs
<|reference_start|>Fractional list packing for layered graphs: The fractional list packing number $\chi_{\ell}^{\bullet}(G)$ 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 $\chi_{\ell}^{\bullet}(G)$, 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 $\chi_{\ell}^{\bullet}(G)$ is bounded from above by the pathwidth of $G$ plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.<|reference_end|>
@article{cambie2024fractional,
    title={Fractional list packing for layered graphs},
    author={Stijn Cambie, Wouter Cames van Batenburg},
    journal={arXiv preprint arXiv:2410.02695},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.02695},
    primaryClass={math.CO cs.DM}
}