素距离图的着色与距离集的分类

COLOURING PRIME DISTANCE GRAPH AND CLASSIFICATION OF DISTANCE SET

设 P 表示全体素数的集合,D(?)P,令 Z(D)表示这样一个图:它的顶点集是全体整数的集合,两个顶点 x 和 y 之间有边连结当且仅当(?)x—y(?)D.文[1]证明了.对任意 D(?)P,Z(D)的色数至多是4,因而可按照 Z(D)的色数把 P 的所有子集 D 分为四类,本文给出了两组新的4类子集簇,并给出若干个 D 是3类子集的充分条件。

Let P denote the ser of all prime numbers,and D(?)P.Let Z(D) denote the graph with the integers as its vertex and the edges between those pairs of integers with absolute differ- ence in the set D.In it is shown that for any D(?)P,the chromatic numder of Z(D) is at most 4,therefore we can allocate the D to four classes,according as chromatic number of Z (D).In this paper,two new families of class 4 sets are given.Some sufficient conditions for D to be class 3 set are also given.