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arXiv · 2111.03933

On Asymptotic Packing of Geometric Graphs

Daniel W. Cranston·Jiaxi Nie·Jacques Verstraëte·Alexandra Wesolek
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Abstract

A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph $G$, we determine whether or not a plane $G$ is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of $K_n$.

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Daniel W. Cranston, Jiaxi Nie, Jacques Verstraëte, Alexandra Wesolek. 2021-11-06. On Asymptotic Packing of Geometric Graphs. https://arxiv.org/abs/2111.03933

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