Spanning disks in triangulations of surfaces
<|reference_start|>Spanning disks in triangulations of surfaces: Given a triangulation $G$ of a surface $\mathbb{D}$, a spanning disk is a disk $\mathbb{D} \subseteq \mathbb{S}$ containing all the vertices of $G$ such that the boundary of $\mathbb{D}$ is a cycle of $G$. In this paper, we consider the question of when a triangulation of a surface contains a spanning disk. We give a very short proof that every triangulation of the torus contains a spanning disk, which strengthens a theorem of Nevo and Tarabykin. For arbitrary surfaces, we prove that triangulations with sufficiently high facewidth always contain spanning disks. Finally, we exhibit triangulations which do not have spanning disks. This shows that a minimum facewidth condition is necessary. Our results are motivated by and have applications for rigidity questions in the plane.<|reference_end|>
@article{clinch2024spanning,
    title={Spanning disks in triangulations of surfaces},
    author={Katie Clinch and Sean Dewar and Niloufar Fuladi and Maximilian Gorsky
  and Tony Huynh and Eleftherios Kastis and Anthony Nixon and Brigitte
  Servatius},
    journal={arXiv preprint arXiv:2410.04450},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.04450},
    primaryClass={math.CO cs.DM}
}