拓扑编码中的图集合着色与超图
Graph Set-colorings And Hypergraphs In Topological Coding
AI总结：
该研究引入与超图相关的集合着色图，通过交图研究超图拓扑结构，得到多种图格，相关技术可转化为多项式算法，旨在应用于同态加密与非对称密码学以抵御量子智能攻击。
摘要：
为了从拓扑编码中生成更复杂的基于数字的字符串，以抵御配备量子计算的智能攻击，并为量子计算时代提供有效的防护技术，我们将引入容许集合着色的集合着色图，这类图具有重要的密码分析意义，尤其与超图相关。我们利用图的集合着色来反映元素的交集，并添加其他约束要求以表达集合（作为超边）之间更多的关联。由于我们试图找到一些基于图论的简单有效技术用于实际应用，我们使用定义在超边集上、容许集合着色的交图来观察超图的拓扑结构，包括字符串型Topcode-矩阵、集合型Topcode-矩阵、图型Topcode-矩阵、超图型Topcode-矩阵、矩阵型Topcode-矩阵等。我们将证明每个连通图都是某个超图的交图，并研究超图的连通性、超图着色、超图同态、超网络、无标度网络生成器，以及顶点为超图的复合超图（用于高维扩展图）的交图。自然地，我们得到了各种图格，例如边重合交图格、顶点重合交图格、边哈密顿图格、超图格和交网络格。本文中的许多技术都可以转化为多项式算法，因为我们的目标是将超图和图集合着色应用于同态加密和非对称密码学。
英文摘要：
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will introduce set-colored graphs admitting set-colorings that has been considerable cryptanalytic significance, and especially related with hypergraphs. We use the set-coloring of graphs to reflect the intersection of elements, and add other constraint requirements to express more connections between sets (as hyperedges). Since we try to find some easy and effective techniques based on graph theory for practical application, we use intersected-graphs admitting set-colorings defined on hyperedge sets to observe topological structures of hypergraphs, string-type Topcode-matrix, set-type Topcode-matrix, graph-type Topcode-matrix, hypergraph-type Topcode-matrix, matrix-type Topcode-matrix \emph{etc}. We will show that each connected graph is the intersected-graph of some hypergraph and investigate hypergraph's connectivity, colorings of hypergraphs, hypergraph homomorphism, hypernetworks, scale-free network generator, compound hypergraphs having their intersected-graphs with vertices to be hypergraphs (for high-dimensional extension diagram). Naturally, we get various graphic lattices, such as edge-coincided intersected-graph lattice, vertex-coincided intersected-graph lattice, edge-hamiltonian graphic lattice, hypergraph lattice and intersected-network lattice. Many techniques in this article can be translated into polynomial algorithms, since we are aiming to apply hypergraphs and graph set-colorings to homomorphic encryption and asymmetric cryptograph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
