Extremal Graphs without Topological Complete Subgraphs

Extremal Graphs without Topological Complete Subgraphs

On Complete Topological Subgraphs of Certain Graphs

Let G he a graph . We say that G contains a complete k-gon if there are !c vertices of G any two of which are connected by an edge, we say that it contains a complete topological k-gon if it contains k vertices any two of which are connected by paths no two of which have a common vertex (except endpoints) . Following G . DIRAC we will denote complete k-gons by -k = and complete topological k-go...

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 ...