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 [ges ] 3 be an integer, c ∈ (0, 1/2) and G be a Kp-free graph on n vertices with e [les ] cn2 edges. There exists a positive constant α = α (c, p) such that:

(a) some [lfloor ]n/2[rfloor ]-subset of V (G) induces at most (c-4 − α) n2 edges (this answers a question of Paul Erdős);

(b) G can be made bipartite by the omission of at most (c-2 − α) n2 edges.