Twins in graphs

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets A , B of vertices twins if they have the same cardinality and induce subgraphs of the same size. Let t ( G ) be the largest k such that G has twins on k vertices each. We provide the bounds on t ( G ) in terms of the number of edges and vertices using discrepancy results for induced subgraphs. In addition, we give conditions under which t ( G ) = | V ( G ) | / 2 and show that if G is a forest then t ( G ) ≥ | V ( G ) | / 2 − 1 .