Six-Rings in Minimal Five-Color Maps
Arthur Bernhart · American Journal of Mathematics · 1947
Introduction. Kempe 1 and Heawood 2 have shown that five colors are sufficient for coloring any map on a sphere. We conjecture the existence of some maps for which five colors are necessary, and seek the simplest example. For this end we here consider only minimal maps, namely those five-color maps such that any map with fewer regions is four-colorable. This paper investigates the structure of minimal maps by a systematic analysis of rings. By a proper n-ring we mean [1] a cycle of n distinct regions, each adjacent to the regions which precede and succeed it in the cyclic order, [2] but to no other region in the cycle, and [31 dividing the rest of the map into two non-empty sides. This definition is equivalent [in minimal maps] to the usage of Birkhoff 3 who introduced the terminology ring of n regions. The first condition conveys the generic meaning of the word ring as used by many four-color writers without a formal definition. The second condition originated with Birkhoff, and corresponds to what Veblen 4 intended by a simple circuit. For n = 1, 2, 3 it adds nothing to the generic meaning, but Birkhoff excluded these cases thereby implying the tbird condition. A ring divides the regions of a map into three mutually exclusive parts: the regions R of the ring itself, the regions I inside the ring, and the regions 0 outside the ring. Ordinarily the terms inside and outside are interchangeable, but whenever the structure of one side is simpler or more fully known we shall refer to it as the inside, achieving thereby an economy of description. In proper rings, I is a proper subset of the map. Whenever we use the term ring in the generic sense with a meaning other than that set forth in the foregoing definition, we shall warn the reader by calling it an improper ring. Thus a single region forms an improper 1-ring, two adjacent regions form an improper 2-ring, the regions meeting at a vertex form an improper 3-ring, and the regions participating in an edge form an improper 4-ring. Conversely, these examples of improper n-rings are the only possibilities for