Convex hulls of dense balanced graphs

For several graph families ℱ, the authors establish asymptotic estimates for the maximum possible number of extreme points that can be in the convex hull of an n-vertex graph in ℱ. In particular, this is done when ℱ is the set of balanced graphs, and when ℱ is the set of balanced graphs with N=N(n) edges. Similar results are obtained for strictly balanced, strongly balanced and strictly strongly balanced graphs.