关于功能性与对称差——两个引人入胜的图参数的界
Bounds on Functionality and Symmetric Difference -- Two Intriguing Graph Parameters
AI总结：
本文研究了图参数功能性与对称差，完全刻画了随机图的功能性并给出了其上下界，同时证明了区间图和圆弧图的对称差的相应界。
摘要：
功能性（$\mathrm{fun}$）是由 Alecu 等人 [JCTB, 2021] 定义的推广图退化性的图参数。他们研究了功能性与许多其他图参数（树宽、团宽、VC维等）的关系。扩展他们的研究，我们完全刻画了随机图 $G(n,p)$ 在所有可能 $p$ 值下的功能性。对于大范围的 $p$，我们提供了匹配的（在常数因子内）上下界。根据我们对 $G(n,p)$ 的界可以得出，最大功能性（约为 $\sqrt{n}$）在 $p \approx 1/\sqrt{n}$ 时取得。我们对此进行了补充，证明每个具有 $n$ 个顶点的图 $G$ 都有 $\mathrm{fun}(G) \le O(\sqrt{ n \ln n})$，并且给出了由射影平面的关联图提供的近乎匹配的 $\Omega(\sqrt{n})$ 下界。先前已知的功能性下界仅与顶点数呈对数关系。\n此外，我们研究了一个相关的图参数对称差（$\mathrm{sd}$），即“最坏可能”导出子图的所有顶点对中 $|N(u) ~Δ~ N(v)|$ 的最小值。Alecu 等人观察到，对于每个图 $G$ 都有 $\mathrm{fun}(G) \le \mathrm{sd}(G)+1$。他们询问区间图的功能性是否有界。最近，Dallard 等人 [RiM, 2024] 给出了肯定回答，并且他们构造了一个满足 $\mathrm{sd}(G) = \Theta(\sqrt[4]{n})$ 的区间图 $G$（尽管他们没有提及显式界），即他们分离了区间图的功能性和对称差。我们证明了区间图的 $\mathrm{sd}$ 至多为 $O(\sqrt[3]{n})$，并提供了另一个满足 $\mathrm{sd}(G) = \Theta(\sqrt[4]{n})$ 的区间图 $G$ 的例子。此外，我们证明了圆弧图的 $\mathrm{sd}$ 为 $\Theta(\sqrt{n})$。
英文摘要：
Functionality ($\mathrm{fun}$) is a graph parameter that generalizes graph degeneracy defined by Alecu et al. [JCTB, 2021]. They research the relation of functionality to many other graphs parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we completely characterize the functionality of random graph $G(n,p)$ for all possible $p$. We provide matching (up to a constant factor) lower and upper bound for a large range of $p$. It follows from our bounds for $G(n,p)$, that the maximum functionality (roughly $\sqrt{n}$) is achieved for $p \approx 1/\sqrt{n}$. We complement this by showing that every graph $G$ on $n$ vertices have $\mathrm{fun}(G) \le O(\sqrt{ n \ln n})$ and we give a nearly matching $Ω(\sqrt{n})$-lower bound provided by incident graphs of projective planes. Previously known lower bounds for functionality were only logarithmic in the number of vertices. Further, we study a related graph parameter symmetric difference ($\mathrm{sd}$), the minimum of $|N(u) ~Δ~ N(v)|$ over all pairs of vertices of the ``worst possible'' induced subgraph. It was observed by Alecu et al. that $\mathrm{fun}(G) \le \mathrm{sd}(G)+1$ for every graph $G$. They asked whether the functionality of interval graphs is bounded. Recently, Dallard et al. [RiM, 2024] answered this positively and they constructed an interval graph $G$ with $\mathrm{sd}(G) = Θ(\sqrt[4]{n})$ (even though they did not mention the explicit bound), i.e., they separate the functionality and symmetric difference of interval graphs. We show that $\mathrm{sd}$ of interval graphs is at most $O(\sqrt[3]{n})$ and we provide a different example of an interval graph $G$ with $\mathrm{sd}(G) = Θ(\sqrt[4]{n})$. Further, we show that $\mathrm{sd}$ of circular arc graphs is $Θ(\sqrt{n})$.
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