The circular embedding conjecture

The circular embedding conjecture

Definitions and mathematical statement

Let $G$ be a finite simple $2$-connected graph, meaning that it has at least three vertices and remains connected after deleting any one vertex. A surface here means a compact connected topological $2$-manifold without boundary. It may be orientable or nonorientable, and its genus is not fixed in advance.

An embedding of $G$ in a surface places vertices at distinct points and realizes edges as arcs whose interiors are mutually disjoint and avoid the vertices. It is cellular if each component of the surface minus the embedded graph is an open disc; these components are its faces. A facial boundary is usually described by a closed walk, which may in general repeat vertices or edges.

The circular embedding conjecture asks whether every such $G$ has a cellular embedding in some surface in which the boundary walk of every face is a simple cycle. Thus each facial boundary visits each of its vertices once before returning to its start; the face closes to a disc bounded by an embedded circle.

The surface and rotation of incident edges may be chosen to suit $G$. No planar embedding or orientable embedding is required. Each edge has two sides in a surface, so facial cycles in such an embedding cover every edge twice when faces are counted with multiplicity. However, the target includes the surface condition at vertices: gluing discs along an arbitrary cycle cover need not give manifold neighbourhoods.

Short English statement

Can every 2-connected graph be embedded on some closed surface so each face is a disc bounded by a simple cycle?

Research attempt: a local criterion for gluing cycle discs

Scope and current literature.

GPT-6 Astra Ultra, 1 October 2026. The surface may be nonorientable; the requirement is that vertex neighbourhoods are discs and facial boundaries are simple cycles. A significant current source is Oum's 2026 exposition [S3] of the announced OpenAI proof of the ordinary cycle double cover theorem. Its introduction distinguishes the stronger embedding problem, which remains a separate problem in general, and explains the cubic consequence. The argument here proves that local connection explicitly and supplies a counterexample to naive gluing in higher degree. No proof of the general embedding conjecture is claimed.

1. The link criterion.

Suppose a connected graph has a list of simple circuits covering each edge exactly twice. Attach a closed disc along each listed circuit, identifying its boundary edges and vertices with the corresponding edges and vertices of the graph. Interior points of a graph edge have two incident half-disc neighbourhoods, so they have manifold neighbourhoods. The possible obstruction is at a vertex $v$.

Construct a multigraph $L_v$ with one vertex for each edge incident with $v$. For each face corner at $v$, join the two incident edges used by that corner. Every vertex of $L_v$ has degree two, because each original edge occurs twice in the cover. Moreover there are no loops: a simple circuit uses two distinct incident edges at a vertex. The neighbourhood of $v$ is a cone on this link. It is a disc exactly when $L_v$ is a single circle, allowing a two-edge circle in the multigraph convention. If every link is connected, the finite quotient is a closed connected surface and the attached disc interiors are its faces. Their boundaries remain the prescribed simple circuits.

2. Why cubic vertices cause no obstruction.

At a cubic vertex, label the incident edges $1,2,3$ and let $a_{ij}$ count corners using edges $i,j$. The double-cover conditions give \[ a_{12}+a_{13}=2,\qquad a_{12}+a_{23}=2,\qquad a_{13}+a_{23}=2. \] Consequently all three corner counts equal one. The link is a triangle and therefore connected. A circuit double cover of a connected cubic graph automatically produces the required surface. In particular, applying the external ordinary double-cover theorem gives the cubic case; that implication is not an independent proof of the theorem.

There is also an entirely explicit subfamily. In a properly three-edge-coloured cubic graph, take the three unions of two colour classes and decompose each into its circuit components. Each edge belongs to two of the unions. The preceding link calculation gives a circular embedding directly. Subdividing any edges preserves such an embedding, because the new degree-two vertices merely subdivide facial boundary arcs and still have circular links.

3. A concrete failure of unrestricted gluing.

On vertices $0,1,2,3,4$, the cycles \[ C=(0,1,2,3,4,0),\qquad D=(0,2,4,1,3,0) \] partition $E(K_5)$. The list $C,C,D,D$ is therefore a circuit double cover. At every vertex, however, its link is the disjoint union of two two-edge circles: one joins the two incident $C$-edges and the other joins the two incident $D$-edges. Thus the glued object is pinched at every graph vertex. Even four-connectivity of the original graph does not make this particular cover a surface embedding. This is an obstruction to the proposed construction, not a claim that $K_5$ lacks a circular embedding.

Verification and first remaining gap.

The script [S4] verifies the two edge-disjoint Hamilton cycles of $K_5$ and the two components in each link. The distinction between a list of circuits and a list of even subgraphs is harmless for ordinary coverage after circuit decomposition, but connected vertex links must be checked afterwards. For vertices of degree at least four, the local parity equations permit several link components. The missing step is selecting or modifying a global circuit cover so all those links become connected simultaneously while each face stays simple. The ordinary double-cover theorem alone does not supply that step.

Final status

UNRESOLVED. A local surface certificate and the cubic reduction are proved, and a higher-degree gluing obstruction is verified. The cubic existence consequence is credited to the current external theorem; this attempt does not resolve arbitrary two-connected graphs. Completion estimate: no defensible global percentage.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3162 (OPG-138), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, The circular embedding conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/the_circular_embedding_conjecture.
• S3 S.-i. Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition, arXiv:2607.16356, version 3 $2026$, especially the introduction and discussion of strong embeddings. Checked 1 October 2026. https://arxiv.org/abs/2607.16356.
• S4 GPT-6 Astra Ultra, exact link and incidence checks, 1 October 2026: scripts/check_attempt_batch02_connectivity_2026_10_01.py in this repository; run with Python 3.