Extremal subgraphs for two graphs

Extremal Subgraphs for Two Graphs

In this paper we study several interrelated extremal graph problems : (i) Given integers n, e, m, what is the largest integer f(n, e, m) such that every graph with n vertices and e edges must have an induced m-vertex subgraph with at least f(n, e, m) edges? (ü) Given integers n, e, e', what is the largest integer g(n, e, e') such that any two n-vertex graphs G and H, with e and e' edges, respec...

Extremal subgraphs of random graphs

Let K` denote the complete graph on ` vertices. We prove that there is a constant c = c(`), such that whenever p ≥ n−c, with probability tending to 1 when n goes to infinity, every maximum K`-free subgraph of the binomial random graph Gn,p is (`− 1)partite. This answers a question of Babai, Simonovits and Spencer [BSS90]. The proof is based on a tool of independent interest: we show, for instan...

Extremal subgraphs of random graphs

We shall prove that if L is a 3-chromatic (so called "forbidden") graph, and -R" is a random graph on n vertices, whose edges are chosen indepen-6" is a bipartite subgraph of R" of maximum size, -F" is an L-free subgraph of R" of maximum size, dently, with probability p, and then (in some sense) F" and 6" are very near to each other: almost surely they have almost the same number of edges, and ...

Extremal Graphs without Topological Complete Subgraphs

Induced subgraphs of Z and related extremal graphs

We prove in a new way that the maximum number of edges an induced subgraph of the 2-grid Z on n vertices is given by E2(n) = ⌊2n − √ 2n⌋. This was first proved by Harary and Harborth in 1976 by induction. Our method suggests some possible ways to obtain analog results in higher dimensions via projections and the Box Theorem by Bollobás and Thomas from 1995. – We discuss some of these key tools ...