Decomposing graphs into edges and triangles

We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\dots ,C_{\mathrm{\ell }}$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le (1/2+o(1))n^2$. This result implies the asymptotic version of the old result of Erd\H{o}s, Goodman and P'osa that asserts the existence of such a decomposition with $\mathrm{\ell }\le n^2\mathrm{/}4$.