Asia Mathematika

An International Journal. ISSN: 2457-0834 (online)

Volume 3, Issue 1, April 2019, Pages: 60-69

On Coloring Distance Graphs

V. Yegnanarayanan

Department of Mathematics, School of Arts, Science and Humanities, SASTRA Deemed to be University, Thanjavur-613401, Tamilnadu, India

Abstract

Some problems become very popular and demand rapt attention while others just collect dust on the shelf. Big Conferences like this are a nice breeding ground for advertising or publicizing both old and new problems. Consider for example the famous unit-distance problem. This problem asks for the smallest number of colors needed to color the points of the plane R2 so that points unit distance apart have distinct colors. This problem was attributed to atleast five different mathematicians in a variety of combinations. Edward Nelson, Hugo Hadwiger, Paul Erdos, Martin Gardner and Leo Moser. It is now known as Hadwiger-Nelson problem (HNP). The problem was actually formulated by Edward Nelson in 1950 when he was a graduate student at the University of Chicago. Nelson called it the alternative four-color problem as it dealt with the plane and four colors. He proved that atleast four colors are needed. John Isbell upon learning about the problem from Nelson, proved in 1951 that the unit distance graph is 7-colorable. This problem might be a typical candidate for a famous problem. It is simple enough for high school student to understand, it is easy to describe and looks deceptively easy to solve. After all, all one needs is to construct a finite set of points in the plane that requires more than 4-colors or prove that any set of points is 4-colorable. As a matter of fact, while hundreds of papers dedicated to variations of this problem have been published, no progress has been made on the actual chromatic number of the unit distance graph since its inception. In this paper this problem is discussed further. Its variations, enormity, challenges, bottleneck and road ahead are quite mind-boggling. However, upon reading this one gets clarity regarding the role of several pure mathematics concepts emanating from set theory, algebra, analysis, topology and number theory. The author of this paper was fascinated very much and is completely engrossed with this problem since 2010. The following are some of his results in this topic.
Given a subset D in Z , an integer distance graph is a graph G(Z; D) with Z as vertex set and with an edge joining two vertices u and v iff ju − vj 2 D. The author considered the problem of determining χ(G(Z; D)) when D is either a) a set of (n + 1) positive integers for which the nth power of the last is the sum of the nth powers of the previous terms or b) a set of Pythagorean quadruples or c) a set of Pythagorean n-tuples or d) a set of square distances or e) a set of abundant numbers or deficient numbers or Carmichael numbers or f) a set of polytope numbers or g) a set of happy numbers or lucky numbers or h) a set of Lucas numbers or i) a set of Ulam numbers or j) a set of weird numbers. In addition, he also found some useful upper and lower bounds for general cases too technical to mention here [14]

Keywords

Graphs, chromatic number, plane-coloring problem

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