On a generalized Erdos-Rademacher problem

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers s, t and an n-vertex graph G with [n(2)/4] + t edges and triangle covering number s, we determine (for large n) sharp bounds on the minimum number of triangles in G and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erdos, and Lovasz-Simonovits whose results apply only to s <= t. Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture.