Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler’s Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap.

💡 Research Summary

The paper presents a fully mechanized proof of a discrete version of the Jordan Curve Theorem (JCT) using the Coq proof assistant and a combinatorial model called hypermaps. A hypermap is defined as a triple (D, α₀, α₁) where D is a finite set of darts and α₀, α₁ are permutations representing 0‑links (edges) and 1‑links (vertices). From these permutations the authors derive orbits that correspond to edges, vertices, faces and connected components. They formalize basic topological invariants: the Euler characteristic χ = v + e + f − d, the genus g = c − χ⁄2, and prove that χ is always even and that for a non‑empty connected planar hypermap (c = 1) the classic Euler formula χ = 2 holds.

The implementation starts with an inductive type dim (zero | one) and a representation of darts as natural numbers. Hypermaps are built as “free maps” using three constructors: V (empty map), I (insertion of a new dart), and L (linking two darts along a given dimension). Preconditions for insertion and linking guarantee that the resulting structure always has open orbits, which can then be closed by auxiliary functions cA and cA1. These closures turn the open permutations into true bijections, enabling the definition of a face‑traversal function F and its closure cF (the combinatorial analogue of the φ‑operator).

A generic module Sigf abstracts the notion of a permutation together with decidable reachability (expo) and equality of components (eqc). Using this abstraction the authors prove that each orbit is periodic, that reachability within an orbit is decidable, and that the numbers of darts, edges, vertices, faces and components can be computed inductively. The genus theorem and Euler formula are then derived by straightforward arithmetic on these counts, all verified inside Coq with the ZArith library.

The central contribution is the definition of a “ring of faces”. A ring is a non‑empty sequence of double‑links (pairs of darts belonging to the same edge) satisfying four properties: distinct edges, continuity (consecutive faces are adjacent via the corresponding double‑link), closure (the last face is adjacent to the first), and simplicity (all faces are distinct). This structure mimics a Jordan curve drawn as a dotted line in a planar subdivision. The authors define a “break along a ring” operation: starting from the original hypermap M, they iteratively swap the 0‑link of each double‑link in the ring, producing a sequence of hypermaps M₀,…,Mₙ, where Mₙ is the broken hypermap M′.

The main theorem (Discrete Jordan Curve Theorem) states that if M is a planar hypermap with c connected components and R is a ring of faces in M, then the broken hypermap M′ has exactly c + 1 components. The proof proceeds by structural induction on the length of the ring, using previously established planarity criteria and connectivity lemmas. The result captures the essential topological fact that cutting a planar subdivision along a simple closed curve increases the number of connected regions by one, but does so entirely within a combinatorial framework that avoids any embedding or metric notions.

Related work is surveyed: the original Jordan theorem (Jordan 1887, Veblen 1905), Tutte’s combinatorial maps (1979), recent formalizations of Euler’s formula and the Four‑Colour Theorem in Coq, Isabelle, and HOL Light, and earlier attempts at formalizing JCT in continuous settings (Mizar, Flyspeck). The authors argue that their contribution is the first fully formalized, discrete JCT proof using hypermaps, and that the hypermap formalism yields simple constructive planarity and connectivity criteria useful for computational geometry and image analysis.

The paper concludes by highlighting the significance of having a machine‑checked proof of a classic topological theorem in a purely combinatorial setting. It opens the way for extending the approach to higher‑dimensional cell complexes, for integrating the criteria into automated geometry processing pipelines, and for further exploring the interplay between formal verification and discrete topology.