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arXiv · 2206.06877

Intrinsically projectively linked graphs

Joel Foisy·Luis Ángel Topete Galván·Evan Knowles·Uriel Alejandro Nolasco·Yuanyuan Shen·Lucy Wickham
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Abstract

A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges exists and identify new minor-minimal IPL graphs by applying $\Delta-Y$ exchanges to $K_{7}-2e$. We prove that for a nonouter-projective-planar graph $G$, $G+\bar{K}_{2}$ is IPL and describe the necessary and sufficient conditions on a projective planar graph $G$ such that $G+\bar{K}_{2}$ is IPL. Lastly, we deduce conditions for $f(G + \bar{K_{2}})$ to have no nonsplit link, where $G$ is projective planar, $\bar{K_{2}} = \{w_{0},w_{1}\}$, and $f(G + \bar{K_{2}})$ is the embedding onto $\mathbb{R}P^{3}$ with $f(G)$ in $z=0$, $w_{0}$ above $z=0$, and $w_{1}$ below $z=0$ such that every edge connecting ${w_{0},w_{1}}$ to $G$ avoids the boundary of the 3-ball, whose antipodal points are identified to obtain projective space.

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Joel Foisy, Luis Ángel Topete Galván, Evan Knowles, Uriel Alejandro Nolasco, Yuanyuan Shen, Lucy Wickham. 2022-06-14. Intrinsically projectively linked graphs. https://arxiv.org/abs/2206.06877

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