Hanani–Tutte Theorems on Surfaces

Definitions and mathematical statement

Let $G$ be a finite graph and $S$ a surface in the intended class. A drawing places vertices at distinct points and edges as arcs with only finitely many transverse interior crossings, no edge interior through a vertex, and no triple crossing, under the usual general-position convention. Two edges are independent, or nonadjacent, if they have no common endpoint. The drawing is independently even if every pair of independent edges has an even number of crossings. An embedding is a drawing without crossings.

The projective-plane statement quoted in the source is that every drawing of a graph not embeddable in ${\mathbb{R}}\mathrm P^2$ has two independent edges crossing an odd number of times. The question asks for the analogous assertion on every other surface $S$: \[ \forall G\ \forall\text{ drawings }D\text{ of }G\text{ on }S, \quad D\text{ independently even}\ \Longrightarrow\ G\text{ embeds in }S. \] This is the strong Hanani--Tutte existence assertion. It does not require the new embedding to preserve the drawing's rotation system, and it imposes no parity condition on adjacent pairs.

The source's all-surfaces question is historical: its supplied assessment reports both affirmative surface cases and higher-genus counterexamples. The draft preserves that distinction instead of presenting the full universal extension as an unresolved true conjecture. The precise class of surfaces, particularly boundary and noncompact cases, is not fixed by the extraction.

Short English statement

Determine on which surfaces an independently even graph drawing guarantees an embedding. The original broad extension question is retained together with the supplied dated report of known positive cases, counterexamples, and remaining cases.

Sources

• Local UnsolvedMath registry, permanent ID $30001053$, source code catalogue entry, "Hanani--Tutte Theorems on Surfaces." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Discrete Geometry (2009), Oberwolfach Reports, pp. 2477--2556, DOI https://doi.org/10.4171/owr/2008/44.
• Supplied "Fulek--Kyncl counterexample": https://arxiv.org/abs/1709.00508.
• Supplied "Hanani--Tutte survey": https://www.combinatorics.org/ojs/index.php/eljc/article/view/v28i1p43.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled partially solved; says the strong assertion is known for the torus and projective plane, counterexamples exist in orientable genus at least four, and orientable genera two and three and other nonorientable surfaces remain unresolved. This is supplied dated curation, not independently verified; no new surface classification is claimed.