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There has been extensive studies on the following question: given k graphs G1, . . . , G k over a common vertex set of size n, what conditions on Gi ensures a 'colorful' copy of H, i.e., a copy of H containing at most one edge from each Gi? A lower bound on i∈[k] e(Gi) enforcing a colorful copy of a given graph H was considered by Keevash, Saks, Sudakov, and Verstraëte. They defined ex k (n, H) to be the maximum total number of edges of the graphs G1, . . . , G k on a common vertex set of size n having no colorful copy of H. They completely determined ex k (n, Kr) for large n by showing that, depending on the value of k, one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for 3-color-critical graphs.We prove their conjecture for 4-color-critical graphs and for almost all r-color-critical graphs when r > 4. Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of k. This answers a question of Keevash, Saks, Sudakov, and Verstraëte.