Sharp bounds for decomposing graphs into edges and triangles

For a real constant $\alpha$, let ${\pi }_3^{\alpha }(G)$ be the minimum of twice the number of $K_2$’s plus $\alpha$ times the number of $K_3$’s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let ${\pi }_3^{\alpha }(n)$ be the maximum of ${\pi }_3^{\alpha }(G)$ over all graphs $G$ with $n$ vertices. The extremal function ${\pi }_3^3(n)$ was first studied by Gy\H{o}ri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315–320]. In a recent progress on this problem, Kr'al’, Lidick'y, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465–472] proved via flag algebras that ${\pi }_3^3(n)\le (1\mathrm{/}2+o(1))n^2$. We extend their result by determining the exact value of ${\pi }_3^{\alpha }(n)$ and the set of extremal graphs for all $\alpha$ and sufficiently large $n$. In particular, we show for $\alpha =3$ that $K_n$ and the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$aretheonlypossibleextremalexamplesforlarge$n$.