Sharp bounds for decomposing graphs into edges and triangles

For a real constant a, let p a 3 (G) be the minimum of twice the number of K2's plus a times the number of K3's over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let p a 3 (n) be the maximum of p a 3 (G) over all graphs G with n vertices. The extremal function p3 3 (n) was first studied by Gy <spacing diaeresis>ori and Tuza (Studia Sci. Math. Hungar. 22 (1987) 315-320). In recent progress on this problem, Kral', Lidicky, Martins and Pehova (Combin. Probab. Comput. 28 (2019) 465-472) proved via flag algebras that p3 3 (n) similar to (1/2+ o(1))n2. We extend their result by determining the exact value of p a 3 (n) and the set of extremal graphs for all a and sufficiently large n. In particular, we show for a = 3 that Kn and the complete bipartite graph K similar to n/2 similar to,similar to n/2 similar to are the only possible extremal examples for large n.