On the number of edges not covered by monodiromatic copies of a matching-critical graph

Given a graph H, let ƒ(n,H) denote the maximum number of edges not contained in any monochromatic copy of H ina 2-edge-coloring of K_n. The Turan number of a graph H, denoted by ex(n,H), is the maximum number of edges in an n-vertex graph which does not contain H as a subgraph. It is easy to see that f (n, H) ≥ex(n, H) for any H and n. We show that this lower bound is tight for matching-critical graphs including Pertersen graph and vertex- disjoint union of copies of cliques with same order.