Large disjoint subgraphs with the same order and size

Large disjoint subgraphs with the same order and size

For a graph G let f(G) be the largest integer k so that there are two vertex-disjoint subgraphs of G, each with k vertices, and that induce the same number of edges. Clearly f(G) ≤ bn/2c 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 polyno...

Disjoint induced subgraphs of the same order and size

For a graph G, let f(G) be the largest integer k such that there are two vertex-disjoint subgraphs of G each on k vertices, both inducing the same number of edges. We prove that f(G) ≥ n/2 − o(n) for every graph G on n vertices. This answers a question of Caro and Yuster.

Disjoint subgraphs of large maximum degree

Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density

It is proved that every graph with edge density at least d + k and sufficiently many vertices contains a k-connected subgraph on at least t (t ≥ k+1) vertices, or s pairwise disjoint subgraphs with edge density at least d. By a classical result of Mader [11] this implies that every graph with edge density at least 3k and sufficiently many vertices contains a k-connected subgraph with at least r...

Induced subgraphs with distinct size or order

We show that for every 0 < < 1/2, there is an n0 = n0( ) such that if n > n0 then every n-vertex graph G of size at least ( n 2 ) and at most (1 − ) ( n 2 ) contains induced k-vertex subgraphs with at least 10−7k different sizes, for every k ≤ n 3 This is best possible, up to a constant factor. This is also a step towards a conjecture by Erdős, Faudree and Sós on the number of distinct pairs (|...