For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α 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 $\pi _3^\alpha (n)$ 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 Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

中文翻译：

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将图形分解为边和三角形的锐界

对于实常数α， 让$\pi _3^\alpha (G)$是两倍数量的最小值ķ2的加号α倍数ķ3的所有边分解G成副本ķ2和ķ3， 在哪里ķr表示完整的图形r顶点。让$\pi _3^\alpha (n)$是最大值$\pi _3^\alpha (G)$在所有图表上G和n顶点。极值函数$\pi _3^3(n)$最早由 Győri 和 Tuza 研究（研究科学。数学。匈牙利。22(1987) 315–320)。在这个问题上的最新进展中，Král'、Lidický、Martins 和 Pehova（结合。概率。计算。28(2019) 465–472) 通过标志代数证明$\pi _3^3(n) \le (1/2 + o(1)){n^2}$. 我们通过确定的确切值来扩展他们的结果$\pi _3^\alpha (n)$以及所有的极值图集α并且足够大n. 特别是，我们展示了α= 3 那ķn和完整的二分图${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$是大的唯一可能的极值示例n.