Radon numbers for trees

Abstract

We consider P3-convexity on graphs, where a set U of vertices in a graph G is convex if every vertex not in U has at most one neighbour in U.Tverberg’s theorem states that every set of (k−1)(d+1)+1 points in Rd can be partitioned into k sets with intersecting convex hulls. As a special case of Eckhoff’s conjecture, we show that a similar result holds for P3-convexity in trees.A set U of vertices in a graph G is free if no vertex of G has more than one neighbour in U. We prove an inequality relating the Radon number for P3-convexity in trees with the size of a maximum free set.