The Euclidean MST-ratio for Bi-colored Lattices
<|reference_start|>The Euclidean MST-ratio for Bi-colored Lattices: Given a finite set, $A \subseteq \mathbb{R}^2$, and a subset, $B \subseteq A$, the \emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A \setminus B$ divided by the length of the minimum spanning tree of $A$.   The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$.   Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$.   By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than $1.25$.<|reference_end|>
@article{di montesano2024the,
    title={The Euclidean MST-ratio for Bi-colored Lattices},
    author={Sebastiano Cultrera di Montesano, Ondv{r}ej Draganov, Herbert
  Edelsbrunner, and Morteza Saghafian},
    journal={arXiv preprint arXiv:2403.10204},
    year={2024},
    doi={10.4230/LIPIcs.GD.2024.3},
    archivePrefix={arXiv},
    eprint={2403.10204},
    primaryClass={cs.CG math.CO}
}