Large disjoint subgraphs with the same order and size

Abstract

For a graph G let f ( G ) be the largest integer k such that there are two vertex-disjoint subgraphs of G , each with k vertices, and that induce the same number of edges. Clearly f ( G ) ≤ ⌊ n / 2 ⌋ but this is not always achievable. Our main result is that for any fixed α > 0 , if G has n vertices and at most n 2 − α edges, then f ( G ) = n / 2 − o ( n ) , which is asymptotically optimal. The proof also yields a polynomial time randomized algorithm. More generally, let t be a fixed nonnegative integer and let H be a fixed graph. Let f H ( G , t ) be the largest integer k such that there are two k -vertex subgraphs of G having at most t vertices in common, that induce the same number of copies of H . We prove that if H has r vertices then f H ( G , t ) = Ω ( n 1 − ( 2 r − 1 ) / ( 2 r + 2 t + 1 ) ) . In particular, there are two subgraphs of the same order Ω ( n 1 / 2 + 1 / ( 8 r − 2 ) ) that induce the same number of copies of H and that have no copy of H in common.