Toward Gr\"unbaum's Conjecture
<|reference_start|>Toward Gr\"unbaum's Conjecture: Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^*$ with edge set $(E(G)-E(T))^*$. Gr\"unbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3.   While Gr\"unbaum's conjecture remains open, Biedl proved that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 5. By using new structural insights into Schnyder woods, we prove that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 4.<|reference_end|>
@article{ortlieb2024toward,
    title={Toward Gr\"unbaum's Conjecture},
    author={Christian Ortlieb, Jens M. Schmidt},
    journal={arXiv preprint arXiv:2402.05681},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.05681},
    primaryClass={cs.DM math.CO}
}