Abstract
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say x y , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations.
Highlights
This article is not about finding an alternative proof of the 4-color theorem; instead, it is an exploratory paper aimed at gaining a deeper understanding of why the 4-color theorem is true, Mathematics 2018, 6, 309; doi:10.3390/math6120309 www.mdpi.com/journal/mathematicsMathematics 2018, 6, 309 what it is about a planar graph that guarantees that it can be 4-colored
We find that each such triangulation has a Birkhoff diamond subgraph with endpoints x and y
We have introduced the key notion of Kempe-locking and have used it to establish the coloring property that a minimum counterexample to the 4-color theorem must satisfy with respect to each of its edges
Summary
We use the standard terminology of graph theory: the basic terms vertex, edge, face, simple graph, planar graph, subgraph, induced subgraph, path, cycle, connectivity, adjacent vertices, order, vertex degree, and minimum degree are all assumed to be known (refer to the introductory text by Chartrand and Zhang [1]). This article is primarily about the coloring of planar graphs. We restrict our consideration to coloring the vertices of simple, connected, planar graphs, and, as is usual, to proper colorings of such graphs, colorings in which no two adjacent vertices have the same color. In studying the 4-color problem, we need consider only planar graphs that are triangulations, graphs in which all faces are delineated by three edges. After properly coloring the triangulation, the inserted edges can be deleted, leaving the original graph with a proper coloring. If all faces of a planar graph except one are triangular, the graph is referred to as a near-triangulation. We use the term configuration for a near-triangulation whose non-triangular face is the infinite face
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