Research on diameter graphs of finite point sets defined by the number of distance

• Main Authors: Xianglin WEI, Yue CONG, Feixing GAO
• Format: Article
• Language: zho
• Published: Hebei University of Science and Technology 2015-04-01
• Series: Journal of Hebei University of Science and Technology
• Subjects: combinatorial mathematics; distinct distance; diameter graph; k-distance sets
• Online Access: http://xuebao.hebust.edu.cn/hbkjdx/ch/reader/create_pdf.aspx?file_no=b201502005&flag=1&journal_

A planar point set X is called a k-distance set if there are exactly k distinct distances defined by every two points in X, and the longest distance is called diameter D. The set of the endpoints of all diameters is denoted by XD . Let m=m(X)=|XD| be the number of elements of XD, and the diameter graph DG(XD) be all diameters in X. There are many results on determining the value of g(k) when k≤6, where g(k) is the number of points of the largest point set having k distinct distances. We consider planar point sets for the case of k≥7. Firstly, we perform an analysis on the degree value d(v) of all vertices in k-distance DG(XD) for m=|XD|=2k-1, and obtain that d(v)≤2. Based on this result, we research the case of 7-distance. We get XD=R15-3 when the 7-distance sets DG(XD)=P10∪P2. The result provides a theoretical foundation for further discussions on 7-distance sets.