A formal proof of the four color theorem

A formal proof has not been found for the four color theorem since 1852 when Francis Guthrie first conjectured the four color theorem. Why? A bad idea, we think, directed people to a rough road. Using a similar method to that for the formal proof of the five color theorem, a formal proof is proposed in this paper of the four color theorem, namely, every planar graph is four-colorable. The formal proof proposed can also be regarded as an algorithm to color a planar graph using four colors so that no two adjacent vertices receive the same color.

💡 Research Summary

The paper under review claims to present a fully formal, non‑computer‑assisted proof of the Four‑Color Theorem (FCT) by adapting the classic inductive argument used for the Five‑Color Theorem. The author, Limin Xiang, begins by recalling the historical background: the conjecture was first stated by Francis Guthrie in 1852, and the only widely accepted proof to date is the computer‑based proof by Appel and Haken (1976), later refined by Robertson, Sanders, Seymour, and Thomas. Because the computer‑based proof relies on exhaustive checking of thousands of reducible configurations, some mathematicians remain uneasy about its “human‑readable” nature. Xiang therefore proposes a new approach that avoids massive case‑checking.

Notation and Preliminary Lemma
The paper introduces standard graph‑theoretic notation: a planar graph G = (V, E), degree d(v) of a vertex v, and the concept of n‑colorability. It defines the operation G – v (removing a vertex and its incident edges) and its inverse G + v. Crucially, the author defines a Kempe chain as a maximal connected subgraph using only two colors i and j, and a Kempe circle as a Kempe chain together with a vertex w that connects to both ends of the chain, forming a closed loop.

Lemma 1 is the well‑known statement that every planar graph contains at least one vertex of degree ≤ 5. The proof uses Euler’s formula and the fact that each face is bounded by at least three edges, yielding an average degree < 6. This lemma guarantees the existence of a “low‑degree” vertex that can be removed in an inductive step.

Structure of the Proof
The main theorem is proved by induction on the number n of vertices. Three broad cases are considered:

1. n ≤ 4 – Trivial, any graph with at most four vertices can be colored with four colors.
2. n = 5 – The maximal planar graph on five vertices (the complete graph K₅ minus one edge) is shown to be 4‑colorable (Fig. 1a). Any planar graph with five vertices is a subgraph of this configuration, so it inherits 4‑colorability.
3. n ≥ 6 – By Lemma 1, there exists a vertex vₙ with degree d(vₙ) ≤ 5. Remove vₙ, obtain G – vₙ, which by the induction hypothesis is 4‑colorable. The rest of the proof is a detailed case analysis of how to extend this coloring to vₙ.

The case analysis for n ≥ 6 is subdivided according to the degree of vₙ and the pattern of colors used on its neighbors:

• Case 3.1: The neighbors of vₙ collectively use at most three distinct colors. Then a fourth unused color can be assigned to vₙ directly.
• Case 3.2: d(vₙ) = 4 and the four neighbors each have a distinct color. The author orders the neighbors clockwise as 1ᵥ, 2ᵥ, 3ᵥ, 4ᵥ with colors c₁,…,c₄. Two sub‑cases are examined:
  • 3.2.1: If vertices 1ᵥ and 3ᵥ are not connected by a (c₁,c₃) Kempe chain, swapping colors along that chain frees color c₃ for vₙ.
  • 3.2.2: If such a chain exists, a (c₂,c₄) Kempe chain is sought, and a Kempe circle is formed to allow a color swap that frees a color for vₙ.
• Case 3.3: d(vₙ) = 5 and the five neighbors use all four colors, with exactly two neighbors sharing a color. Two configurations are distinguished:
  • 3.3.1 (the duplicated color vertices are adjacent in the cyclic order). This splits further into sub‑cases depending on the presence of (c₁,c₃) or (c₁,c₅) Kempe chains, leading to appropriate swaps.
  • 3.3.2 (the duplicated color vertices are separated by other colors). This leads to a more intricate cascade of Kempe chain searches: (c₅,c₂), then possibly (c₄,c₂), then (c₄,c₁), and finally (c₅,c₂) again, each time forming a Kempe circle and swapping colors to eventually free a color for vₙ.

Each sub‑case is illustrated by a figure (Fig. 1b–j). The author claims that in every possible configuration one of the described Kempe‑chain manipulations succeeds, thereby extending the 4‑coloring to the original graph G.

Algorithmic Interpretation
Section 4 presents a hand‑checked flow chart (Fig. 2) that encodes the decision tree described above. The author argues that this flow chart constitutes an explicit algorithm: given any planar graph, repeatedly remove a low‑degree vertex, color the reduced graph recursively, then re‑insert the vertex and apply the appropriate Kempe‑chain recoloring step as dictated by the flow chart.

Conclusions and Critical Assessment
The paper concludes that the Four‑Color Theorem can be proved directly without computer assistance, using a method analogous to the five‑color proof. It emphasizes that the flow chart provides a constructive coloring algorithm.

From a critical standpoint, the proof reproduces the classic Kempe‑chain technique that was historically proposed by Alfred Kempe in 1879 and later shown to be incomplete by Heawood (1890). The central flaw in Kempe’s original argument is precisely the assumption that, when a Kempe chain exists, swapping colors along it will always free a color for the re‑inserted vertex. Counterexamples (e.g., certain configurations of a planar graph with a vertex of degree 5) demonstrate that both relevant Kempe chains can exist simultaneously, preventing any simple swap from resolving the conflict. Xiang’s paper attempts to address this by introducing additional sub‑cases and more elaborate chains, but the analysis does not rigorously prove that at least one of the required Kempe chains is absent in every possible configuration. The case distinctions rely heavily on the planar embedding and on the assumption that certain vertices are “adjacent in clockwise order,” yet no exhaustive combinatorial argument is supplied to guarantee coverage of all planar embeddings.

Moreover, the proof lacks a formal verification component. Modern computer‑assisted proofs of the Four‑Color Theorem (Appel‑Haken, Robertson‑Sanders‑Seymour‑Thomas) are built on a reducibility argument that has been checked by independent programs and later formalized in proof assistants (Coq, HOL Light). Xiang’s approach, while algorithmic in spirit, does not provide a mechanized verification that the flow chart indeed terminates with a proper coloring for every planar graph. The reliance on hand‑drawn figures (Fig. 1a–j) and a “hand‑checked” flow chart leaves the proof vulnerable to overlooked configurations.

In summary, the paper offers an interesting pedagogical exposition of Kempe‑chain recoloring and attempts to systematize it into an algorithmic proof. However, the core logical gap—ensuring the existence of a suitable Kempe chain or the impossibility of simultaneous obstructing chains—is not resolved. Consequently, the claimed “formal proof” does not meet the rigorous standards required for a definitive proof of the Four‑Color Theorem, and the mathematical community continues to regard the computer‑assisted proofs as the accepted resolution of the problem.