## Diophantine approximations and its applications to graph colouring problems (1998)

### BibTeX

@MISC{Zhu98diophantineapproximations,

author = {Xuding Zhu},

title = {Diophantine approximations and its applications to graph colouring problems},

year = {1998}

}

### OpenURL

### Abstract

For a real number x, let ||x| | denote the distance from x to the nearest integer. Suppose x1 < x2 < x3 are positive integers with gcd(x1, x2, x3) = 1. This paper proves the following: if (x1, x2, x3) �= (1, 2, 3s) for an integer s and x3 � = x1 +x2, or x3 = x1 +x2 but x1 ≡ x2 (mod 3), then there is a real number t such that ||txi| | ≥ 1/3 (for i = 1, 2, 3). If (x1, x2, x3) = (1, 2, 3s) or x3 = x1 + x2 and x1 � ≡ x2 (mod 3), then no such t exists, i.e., for any t, there is an i such that ||txi| | < 1/3. This result is connected to problems of different fields of mathematics. Firstly, it is a strengthening of the k = 3 case of Wills’ conjecture, which says that for any k positive integers x1, x2, · · · , xk, there is a real number t such that ||txi| | ≥ 1 k+1. Secondly, it is ap-plied to graph theory in determining the chromatic number of certain distance graphs, which confirms a conjecture proposed independently by Chen, Chang and Huang [J. Graph Theory, 25(1997)287-294] and Voigt [Ars Combinatoria, to appear]. Thirdly, it has an application to the so called view obstruction problem in the 3 dimensional Euclidean space. Fourthly, it has an application to the study of flows in graphs and matroids.