AI 中文总结
本文针对经典图性质难以推广到量子图的问题，提出量子集合子集的新定义，统一量子图性质的经典推广定义，得到着色等已有概念并改进独立集等概念。
AI 中文摘要
经典图的许多性质是基于顶点集的子集定义的，例如连通分支是满足$X \times X^c$与边集$E(G)$不相交的子集$X \subseteq V(G)$，独立集是满足$X \times X$与$E(G)$不相交的子集。将这些定义直接推广到量子图中存在困难，因为量子集合的子集的自然概念过于僵化，导致将经典性质推广到量子场景的方法五花八门，同一概念存在多个不等价的定义。本文引入了一种自然且动机充分的量子集合子集的替代定义，在此基础上提出了量子图性质的统一定义，作为经典定义的直接推广，由此得到了已有的着色和连通分支概念；对于独立集和团，本文的方法提出了与现有定义不同的变体，解决了部分反直觉性质，同时也展示了如何在该框架中恢复重要的现有独立集定义。
英文摘要
Many properties of classical graphs are defined in terms of subsets of the vertex set. Examples include connected components, which are subsets $X \subseteq V(G)$ such that $X \times X^c$ and $E(G)$ are disjoint, or independent sets, for which $X \times X$ and $E(G)$ are disjoint. Direct generalisations of these definitions to quantum graphs are difficult to achieve, since the natural notion of subsets of a quantum set is much too rigid. As a consequence, approaches to generalising these classical properties to the quantum setting have been eclectic. In some cases, multiple inequivalent definitions of the same notion are in use. We introduce a natural and well-motivated alternative definition of subsets of a quantum set. Building on this, we propose unified definitions of quantum graph properties as straightforward generalisations of the classical definitions. We recover this way the established notions of colourings and connected components. For independent sets and cliques, our approach suggests variations that diverge from existing definitions, but address some of their counterintuitive properties. We nevertheless show how to recover an important existing definition of independent sets in our framework.
