Journal ArticleSharp bounds for decomposing graphs into edges and trianglesBlumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, JannullAbstract For a real constant α , let $\pi _3^\alpha (G)$ be the minimum of twice the number of K 2 ’s plus α 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 Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n .2021-03-0110274261Combinatorics, Probability and Computing302271 to 2870963-5483https://doi.org/10.1017/S09635483200003581855653; 1855622National Science Foundation