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 n2 - 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 fH (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 fH (G, t) = Ω (n1 - (2 r - 1) / (2 r + 2 t + 1)). In particular, there are two subgraphs of the same order Ω (n1 / 2 + 1 / (8 r - 2)) that induce the same number of copies of H and that have no copy of H in common.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics