A Direct Proof of the Strong Hanani-Tutte Theorem on the Projective Plane

Abstract

We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives hope to extend it to other surfaces. Moreover, our approach can be used to provide an efficient algorithm turning a Hanani-Tutte drawing on the projective plane into an embedding.

Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4

“…and {B3, D3} cross once, but they are all adjacent with a common vertex outside W , the pairs {C1, B3} and {D2, B3} cross twice, and no other pair of edges has a common crossing. To verify condition 2), we use the fact that every vertex of W has the same rotation in D; namely, (1,2,3). Let E be an embedding of G on an orientable surface S such that the rotation of every vertex from W is (1,2,3).…”

“…To verify condition 2), we use the fact that every vertex of W has the same rotation in D; namely, (1,2,3). Let E be an embedding of G on an orientable surface S such that the rotation of every vertex from W is (1,2,3). Assume without loss of generality that S has minimum possible genus, which implies that E is a 2-cell embedding.…”

“…We draw the black, red and blue edges of G ′ along the edges of K incident with vertices 1, 2 and 3, respectively, so that they reach the special 4-cycles in H. Then we extend these twelve edges using vertex-disjoint paths in H, until they reach the boundary of D ′ . We draw the last portions of these edges inside D ′ , without having to use the embedding K. Due to the planarity of H, the cyclic orders of the colored vertices on the boundaries of the special 4-cycles are "linked" by the black, red and blue edges to opposite cyclic orders around the vertices of W ′ inside D ′ ; in particular, the rotation of each vertex of W ′ in the constructed drawing contains the cyclic subsequence (1,2,3). Moreover, we can make sure that the black and blue edges, incident to the vertices 1 and 3, respectively, are "accessible" from the boundary of H, while the red edges are "hidden".…”

“…Pelsmajer, Schaefer and Stasi [11] extended the strong Hanani-Tutte theorem to the projective plane, using the list of forbidden minors. Colin de Verdière et al [3] recently provided an alternative proof, which does not rely on the list of forbidden minors.…”

“…Theorem 5 (The (strong) Hanani-Tutte theorem on the projective plane [3,11]). If a graph G has a drawing on the projective plane such that every pair of independent edges crosses an even number of times, then G has an embedding on the projective plane.…”