Densidade local em grafos

Abstract

We consider the following problem. Fixed a graph H and a real number α ∈ (0, 1], determine the smallest β = β(α, H) satisfying the following property: if G is a graph of order n such that every subset of αn vertices spans more that βn 2 edges then G contains H as a subgraph. This problem was initiated and motivated by Erdős who conjectured that every triangle-free graph of order n contains a subset of n/2 vertices that spans at most n 2 /50 edges. Our main result shows that i) every triangle-and pentagon-free graph of order n contains a subset of n/2 vertices inducing at most n 2 /64 edges and, ii) if G is a triangle-free regular graph of order n with degree exceeding n/3 then G contains a subset of n/2 vertices inducing at most n 2 /50 edges. Furthermore, if G is not 3-chromatic then G contains a subset of n/2 vertices inducing less than n 2 /54 edges. As a by-product and confirming a conjecture of Erdős asymptotically, we obtain that every n-vertex triangle-free regular graph with degree exceeding n/3 can be made bipartite by removing at most (1/25 + o(1))n 2 edges. We also provide a counterexample to a conjecture of Erdős, Faudree, Rousseau and Schelp.