The Four-Colour Theorem

Abstract

dedicated to professor w. t. tutte on the occasion of his eightieth birthdayThe four-colour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Haken's in several respects.

Approximate min–max relations for odd cycles in planar graphs

“…Below, we give a concise proof. As with the proof of Král and Voss, our proof relies on the Four Colour Theorem [1,11].…”

Efficient Loopy Belief Propagation Using the Four Color Theorem

“…As a matter of fact, that theory also states that at most 6 colors are required to color any map on the plane or the sphere [8]. Of course, and as repeatedly mentioned in the paper, the Four Color Theorem [17] demonstrates that the necessary number of colors is 4. Yet, such coloring is NP-complete and using more colors relaxes the situation, allowing for polynomial time complexity colorings.…”

“…This representation corresponds to a planar graph, that is, a graph that can be drawn in the plane without edge crossings. According to the FCT [17] the chromatic number of the graph is 4 -at most 4 different colors are needed such that no adjacent vertices have the same color. For vertex coloring any map on a torus maximum 7 colors are necessary.…”

“…These placeholder labels can then be replaced by the full set of actual labels. Since image segments form a planar graph, they therefore require at most four placeholder labels by virtue of the Four Color Theorem (FCT) [17] to still have different colors for all adjacent segments. A joint optimization process provides a fast segmentation through the placeholder labels and a fine-grained labeling through the actual labels.…”

Towards an Algorithmic Guide to Spiral Galaxies

“…Given an instance of Spiral Galaxies, any solution can be interpreted as a two-dimensional map, where each galaxy g (and the fields it contains) is a region, and where two distinct regions are adjacent when they contain adjacent fields. The famous four color theorem (see e.g., [8]) tells us that such a map can be colored with at most four colors. The exact algorithm that follows from the above argument can be described easily as follows: generate every possible four-coloring C U of the fields of U ; for each such C U , check whether (a) each connected set of fields of the same color contains a unique galaxy center, and if so, whether (b) it is a valid galaxy, i.e., it satisfies the symmetry condition and is hole-free.…”