Covering the edges of a graph with triangles

In a graph G, let rho(Delta)(G) denote the minimum size of a set of edges and triangles that cover all edges of G, and let alpha(1)(G) be the maximum size of an edge set that contains at most one edge from each triangle. Motivated by a question of Erdos, Gallai, and Tuza, we study the relationship between rho(Delta)(G) and alpha(1)(G) and establish a sharp upper bound on rho(Delta)(G). We also prove Nordhaus-Gaddum-type inequalities for the considered invariants. (c) 2024 The Author(s).