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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 ⌊n2∕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, Erdős, and Lovász–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. [ABSTRACT FROM AUTHOR]

A conjecture of Chung and Graham states that every K4‐free graph on n vertices contains a vertex set of size ⌊n∕2⌋ that spans at most n2∕18 edges. We make the first step toward this conjecture by showing that it holds for all regular graphs. [ABSTRACT FROM AUTHOR]