On the Edge Distribution of a Graph


V. Nikiforov


We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions. Let p ≥ 3 be an integer, c ∈ (0, 1/2) and G be a Kp-free graph on n vertices with e ≥ cn2 edges. There exists a positive constant α = α (c, p) such that : (a) some [n/2]-subset of V(G) induces at most (c/4 - α) n2 edges (this answers a question of Paul Erdos); (b) G can be made bipartite by the omission of at most (c/2 - α) n2 edges.

Publication Title

Combinatorics Probability and Computing