Extending drawings of complete graphs into arrangements of pseudocircles
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Motivated by the successful application of geometry to proving the Harary-Hill 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 𝕊^2 in which the vertices of G are represented as points—no three on a great circle—and the edges of G are shortest-arcs in 𝕊^2 connecting pairs of vertices. Such a drawing has three properties: every edge e is contained in a simple closed curve γ_e such that the only vertices in γ_e are the ends of e; if e f, then γ_e∩γ_f has precisely two crossings; and if e f, then e intersects γ_f at most once, either a crossing or an end of e. We use these three properties to define a pseudospherical drawing of G. Our main result is that, for the complete graph, these three properties are equivalent to the same three properties but with "exactly two crossings" 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.
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