Trees and co-trees in planar 3-connected graphs An easier proof via Schnyder woods
<|reference_start|>Trees and co-trees in planar 3-connected graphs An easier proof via Schnyder woods: Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a spanning tree with maximum degree $k$. In 1970, Gr\"unbaum conjectured that every 3-connected planar graph contains a 3-tree whose co-tree is also a 3-tree. In 2014, Biedl showed that every such graph contains a 5-tree whose co-tree is a 5-tree. In this paper, we present an easier proof of Biedl's result<|reference_end|>
@article{ortlieb2024trees,
    title={Trees and co-trees in planar 3-connected graphs An easier proof via
  Schnyder woods},
    author={Christian Ortlieb, Jens M. Schmidt},
    journal={arXiv preprint arXiv:2402.01230},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.01230},
    primaryClass={cs.DM math.CO}
}