Covering the edges of a graph with triangles

In a graph G, let ${\rho }_{△}\left(G\right)$ denote the minimum size of a set of edges and triangles that cover all edges of G, and let ${\alpha }_1\left(G\right)$ be the maximum size of an edge set that contains at most one edge from each triangle. Motivated by a question of Erdős, Gallai, and Tuza, we study the relationship between ${\rho }_{△}\left(G\right)$ and ${\alpha }_1\left(G\right)$ and establish a sharp upper bound on ${\rho }_{△}\left(G\right)$. We also prove Nordhaus-Gaddum-type inequalities for the considered invariants.