论文标题
混合图的Turán问题
Turán Problems for Mixed Graphs
论文作者
论文摘要
我们研究了混合图的天然Turán问题，即边缘可以指向或无方向性的图形的概括。我们研究一个天然\ textit {turán密度系数}，该{turán密度系数}，该}衡量了一小部分的指向边缘可以具有$ f $的混合图；我们建立了Erdős-Stone-Simonove的类似物，并给出了任何混合图的Turán密度系数（以及相关的极端$ F $ f $ - fo $ family family）的变异表征。这种表征使我们能够强调经典的极端数量与Turán密度系数之间的重要差异。我们表明，Turán密度系数可能是不合理的，但始终是代数。对于每个正整数$ k $，我们都建立了一个混合图的家族，其Turán密度系数具有代数$ K $。
We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Turán density coefficient} that measures how large a fraction of directed edges an $F$-free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán density coefficient of any mixed graph (along with an associated extremal $F$-free family). This characterization enables us to highlight an important divergence between classical extremal numbers and the Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer $k$, we construct a family of mixed graphs whose Turán density coefficient has algebraic degree $k$.
