A Turán-type Extremal Theory of Convex Geometric Graphs

Abstract

We study Turán-type extremal questions for graphs with an additional cyclic ordering of the vertices, i.e. for convex geometric graphs. If a suitably defined chromatic number of the excluded subgraph is bigger than two then the results on convex geometric graphs resemble very much the classical results from the Tur´an theory. On the other hand, in the bipartite case we show some surprising differences, in particular for trees and forests. For example, the Turán function of some convex geometric forests is of the order Θ(n log n), a growth rate that does not occur in the graph Turán theory. We also obtain still another proof of Füredi’s O(n log n) bound on the number of unit distances in a convex n-gon, together with a lower bound showing the limits of this model. The exact growth of the Turán function for several infinite classes of convex geometric graphs is also determined.