Date of Award
Senior Scholars Paper (Open Access)
Colby College. Mathematics and Statistics Dept.
Small, Donald B.
An intriguing concept in convexity theory is the characterization of sets having property Pm,n. A set is said to satisfy property Pm,n if for any collection of m arbitrary points at least n of the (m, 2) line segments determined by the points is contained in the given set. Valentine did the initial work with property P3,1, showing that a set with this property could be expressed as the union of 3 or fewer convex sets. This work was extended by Kay and Guay, who worked with the more general property Pm,n _ Convex: sets and graph theory were linked by Guay, and I have sought to continue this work by combining property P m,n and graph theory. A graph G is said to have property P m,n if given any m arbitrary points in G, there are at least n edges joining these points, n < m. The aim of this paper is to present some results which were obtained while seeking to characterize graphs having property P m,n The maximum diameter of a graph having property P m,n, and the minimum number of edges necessary for property P3,1 have been found. In any graph G, property P m,n can be extended to property P m+1, n+1. Finally, a graph can be generated having no star-complete points. There seem to be several other methods for characterizing graphs having property Pm,n, but I did not have time to study them.
Convex domains, Graph theory
Recommended CitationGaston, Nancy, "On property Pm,n and some applications to graph theory" (1971). Senior Scholar Papers. Paper 92.
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