AI 中文总结
本文研究诱导与非诱导覆盖数之间的有界性，针对不同性质的图类，在240种组合中为219种确定了绑定函数的存在性，并证明遗传客类在三个覆盖数上总有绑定函数。
AI 中文摘要
存在四个覆盖数 $\mathrm{c}_g^{\mathcal{G}}(H)$、$\mathrm{c}_u^{\mathcal{G}}(H)$、$\mathrm{c}_l^{\mathcal{G}}(H)$、$\mathrm{c}_f^{\mathcal{G}}(H)$，每个都以略微不同的方式衡量图 $H$（称为宿主）的边能被某类图 $\mathcal{G}$（称为客类）中的图覆盖的程度。如果我们要求 $\mathcal{G}$ 中的图对应于 $H$ 的诱导子图，则对每个覆盖数 $\mathrm{c}_x^{\mathcal{G}}$ 得到诱导变体 $\mathrm{ic}_x^{\mathcal{G}}$，且对每个图 $H$ 满足 $\mathrm{c}_x^{\mathcal{G}}(H) \leq \mathrm{ic}_x^{\mathcal{G}}(H)$。然而，一般而言，$\mathrm{ic}_x^{\mathcal{G}}$ 不能由 $\mathrm{c}_x^{\mathcal{G}}$ 来界定。如果对于客类 $\mathcal{G}$ 和宿主类 $\mathcal{H}$，存在一个函数 $f$，使得对每个图 $H \in \mathcal{H}$ 有 $\mathrm{ic}_x^{\mathcal{G}}(H) \leq f(\mathrm{c}_x^{\mathcal{G}}(H))$，则称 $f$ 为绑定函数。在本工作中，我们研究客类 $\mathcal{G}$ 和宿主类 $\mathcal{H}$ 的哪些结构性质下存在这样的绑定函数。我们考虑客类 $\mathcal{G}$ 是单调的、遗传的、分量封闭的或都不是，并且具有有界最大平均度、有界色数或都不是。我们考虑的宿主类 $\mathcal{H}$ 具有有界树宽、排除某个子式、有界最大平均度、有界色数或这些性质都没有。对于 $\mathcal{G}$ 和 $\mathcal{H}$ 的性质及覆盖数的 $240$ 个可能的 $3$ 元组中的 $219$ 个，我们要么提供绑定函数，要么提供不存在此类函数的例子。特别地，我们表明对于具有有界最大平均度的遗传客类 $\mathcal{G}$，四个覆盖数中的三个总是存在这样的绑定函数，但第四个可能不存在。
英文摘要
There are four covering numbers $\mathrm{c}_g^{\mathcal{G}}(H),\mathrm{c}_u^{\mathcal{G}}(H),\mathrm{c}_l^{\mathcal{G}}(H),\mathrm{c}_f^{\mathcal{G}}(H)$, each of which measures in a slightly different way how well the edges of a graph $H$ (called a host) can be covered with graphs of a class $\mathcal{G}$ (called a guest class). If we require the graphs of $\mathcal{G}$ to correspond to induced subgraphs of $H$, we obtain an induced variant $\mathrm{ic}_x^{\mathcal{G}}$ for each covering number $\mathrm{c}_x^{\mathcal{G}}$ which satisfies $\mathrm{c}_x^{\mathcal{G}}(H) \leq \mathrm{ic}_x^{\mathcal{G}}(H)$ for every graph $H$. Yet, in general $\mathrm{ic}_x^{\mathcal{G}}$ cannot be bounded in terms of $\mathrm{c}_x^{\mathcal{G}}$. If there exists for a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ a function $f$ such that $\mathrm{ic}_x^{\mathcal{G}}(H) \leq f(\mathrm{c}_x^{\mathcal{G}}(H))$ for every graph $H \in \mathcal{H}$, we call $f$ a binding function. Within this work, we study for which structural properties of a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ such binding functions exist. We consider guest classes $\mathcal{G}$ that are monotone, hereditary, component-closed or neither, and have bounded maximum average degree, bounded chromatic number or neither. The host classes $\mathcal{H}$ we consider have bounded treewidth, exclude some minor, have bounded maximum average degree, bounded chromatic number, or none of these properties. For $219$ out of the $240$ possible $3$-tuples of properties for $\mathcal{G}$ and $\mathcal{H}$ and covering numbers we either provide a binding function or an example where no such function exists. In particular, we show that such binding functions always exist for hereditary guest classes $\mathcal{G}$ of bounded maximum average degree for three of the four covering numbers, but may not for the fourth kind.
