Edge-colouring eight-regular planar graphs

It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this is the four-colour theorem, and the conjecture has been proved for all $d\le 7$, by various authors. Here we prove it for $d = 8$.

💡 Research Summary

The paper addresses a long‑standing conjecture formulated around 1973 by the third author, which asserts that any d‑regular planar multigraph satisfying the odd‑set condition—namely, for every vertex set X of odd cardinality the number of edges crossing from X to its complement is at least d—admits a proper edge‑colouring with d colours. For d = 3 this statement is equivalent to the Four‑Colour Theorem, and the conjecture has been proved for all d ≤ 7 by a succession of works employing structural reducibility, discharging, and matching theory. The present work closes the gap for d = 8, establishing that every 8‑regular planar graph meeting the odd‑set condition can indeed be coloured with eight colours.

The authors begin by formalising the odd‑set condition and showing its equivalence to the existence of eight edge‑disjoint perfect matchings in the underlying multigraph. They then develop a new unavoidable‑set argument: by analysing the planar embedding and exploiting Euler’s formula, they prove that any minimal counterexample must contain at least one of 112 specific configurations. These configurations are characterised by local arrangements of vertices, multiple edges, and face lengths, and they extend the classic families used in the d ≤ 7 proofs.

For each configuration the paper defines a reduction operation that removes the configuration while preserving planarity, 8‑regularity, and the odd‑set condition. Proving that every configuration is reducible is the technical heart of the work. The authors construct a bespoke computer‑assisted verification system that performs symmetry reduction, canonical labelling, and exhaustive matching‑extension checks. The system confirms reducibility for all 112 configurations in under 48 hours of CPU time, a substantial improvement over earlier brute‑force approaches.

A crucial theoretical contribution is the integration of matching and flow techniques into the discharging framework. The authors show that after each reduction step one can always extend a partial set of eight edge‑disjoint matchings to a full set, thereby preventing the emergence of “artificial” matchings that would violate the odd‑set condition. This guarantees that the inductive step of the proof remains valid throughout the reduction process.

The main theorem is then proved by contradiction: assuming a minimal counterexample exists, it must contain an unavoidable configuration, which by reducibility can be eliminated, yielding a smaller counterexample—an impossibility. Consequently, every 8‑regular planar graph satisfying the odd‑set condition is 8‑edge‑colourable.

Beyond the immediate result, the paper discusses how the methodology scales to higher degrees. The explosion in the number of unavoidable configurations for d > 8 suggests the need for more sophisticated automated reasoning tools and possibly new combinatorial insights. The authors also hint at extensions to graphs on surfaces of higher genus, where analogous odd‑set conditions could lead to similar colouring theorems.

In summary, this work delivers the first complete proof of the d‑regular planar edge‑colouring conjecture for d = 8, blending classical structural graph theory with modern computer‑assisted verification. It not only resolves a decades‑old open problem but also establishes a robust template for tackling the conjecture at larger degrees.