Extending drawings of complete graphs into arrangements of pseudocircles
Abstract
Motivated by the successful application of geometry to proving the HararyHill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points  no three on a great circle  and the edges of $G$ are shortestarcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\gamma_e$ such that the only vertices in $\gamma_e$ are the ends of $e$; (2) if $e\ne f$, then $\gamma_e\cap\gamma_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $\gamma_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)(3) are equivalent to the same three properties but with "precisely two crossings" in (2) replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.06053
 Bibcode:
 2020arXiv200106053A
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Geometry;
 05C10;
 52C10;
 52C30