Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph

The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976 with electronic computer that Appel and Haken first gave a proof by finding and verifying 1936 reducible unavoidable sets, and a simplified proof of Robertson, Sanders, Seymour and Thomas in 1997 only involved 633 reducible unavoidable sets, both proofs could not be realized effectively by hand. Until now, finding the reducible unavoidable sets remains the only successful method to use, which came from Kempe’s first “proof” of the four color problem in 1879. An alternative method only involving 4 reducible unavoidable sets for proving the four color theorem is used in this paper, which takes form of mathematical proof rather than a computer-assisted proof and proves both the four color conjecture and the uniquely 4-colorable planar graph conjecture by mathematical method.

💡 Research Summary

This paper presents a claimed mathematical proof of two long-standing conjectures: the Four Color Theorem and the conjecture on uniquely 4-colorable planar graphs. The author asserts a departure from the computer-assisted proofs of Appel-Haken (1976) and Robertson-Sanders-Seymour-Thomas (1997), which relied on checking thousands of reducible unavoidable configurations. Instead, the paper proposes a pure mathematical method requiring only four reducible unavoidable sets.

The core methodology rests on three pillars. First, it develops a color-coordinate system theory for graphs. This theory classifies k-colorable graphs into three categories: uniquely k-colorable, quasi-uniquely k-colorable, and pseudo-uniquely k-colorable. The paper then delineates the characteristics of maximal planar graphs belonging to each of these categories for k=4.

Second, it establishes a generating operation system for maximal planar graphs. Key operations are “extending” and “contracting,” particularly the k-wheel operations. This system shows how any maximal planar graph can be constructed from a lower-order one via wheel extension, providing a foundation for inductive proofs.

Third, it leverages the chromatic polynomial f(G, t). The author notes that proving f(G, 4) > 0 for any maximal planar graph G is equivalent to proving the Four Color Theorem. The proof strategy involves deriving recurrence formulas for the chromatic polynomial under contracting 4-wheel and 5-wheel operations and proceeding by induction.

A significant intermediate result is the resolution of the Frioini-Wilson-Fisk Conjecture (also known as the Jensen-Toft Conjecture) concerning uniquely 4-colorable planar graphs. The paper proves that “a 4-chromatic maximal planar graph G is uniquely 4-colorable if and only if G is a recursive maximal planar graph” (specifically termed a (2,2)-FWF graph in the paper). This provides a complete structural characterization.

Synthesizing these components—the operation system, the color-coordinate classification, and the analysis of chromatic polynomials—the paper culminates in a proof by mathematical induction that for any maximal planar graph G, f(G, 4) > 0. This final statement directly implies the truth of the Four Color Theorem. The author emphasizes that this proof is intended to be verifiable by hand, in contrast to the computer-dependent nature of all previous acknowledged proofs, thereby offering a novel and purely mathematical approach to these classic problems in graph theory.