Sparse graphs with local covering conditions on edges

In 1988, Erd\H{o}s suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In this paper, we study a natural generalization of the question of Erd\H{o}s in which we replace `triangle' with `clique of order $k$' for ${k\ge 3}$. We completely resolve this generalized question with the characterization of all extremal graphs. Motivated by applications in data science, we also study another generalization of the question of Erd\H{o}s where every edge is required to be in at least $\ell$ triangles for $\ell\ge 2$ instead of only one triangle. We completely resolve this problem for $\ell = 2$.