The Birkhoff Diamond as Double Agent

TL;DR

This paper investigates the role of the Birkhoff diamond in 4-colorability of planar graphs, suggesting it is essential for Kempe-locking and influences the structure of potential minimal counterexamples.

Contribution

It introduces the novel idea that the Birkhoff diamond is necessary for Kempe-locking, implying a dual role in the 4-color theorem context.

Findings

Birkhoff diamond found in all Kempe-locked triangulations

Heuristic supports the necessity of Birkhoff diamond for Kempe-locking

Proposes Birkhoff diamond influences minimal counterexamples

Abstract

Despite the existence of a proof of the 4-color theorem, it would seem that there is still more to learn about why any planar graph is 4-colorable. To that end, we take another look at the Birkhoff diamond and discover something new and intriguing: after an extensive search for (rare) Kempe-locked triangulations, we find a Birkhoff diamond subgraph in each one. We offer a heuristic argument as to why that result is not only reasonable but also to be expected and posit that the presence of a Birkhoff diamond is necessary to Kempe-locking. If that conjecture is true, it means that the Birkhoff diamond plays a double role in the matter of 4-colorability, simultaneously working for opposite sides of whether a given planar graph could possibly be a minimum counterexample.