Multiple Kronecker Covering Graphs
Tomaz Pisanski, Wilfried Imrich
一个图可以以不止一种方式是克罗内克覆盖。在本文中，我们将探讨这一现象。使用这种方法，我们证明了两个图的最小公共覆盖不需要是唯一的。
A graph may be the Kronecker cover in more than one way. In this note we explore this phenomenon. Using this approach we show that the least common cover of two graphs need not be unique.
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The neighborhood complex $N(G)$ is a simplicial complex assigned to a graph $G$ whose connectivity gives a lower bound for the chromatic number of $G$. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers $m$ and $n$ greater than 2, we construct connected graphs $G$ and $H$ such that $N(G) \cong N(H)$ but $χ(G) = m$ and $χ(H) = n$. We also construct a graph $KG_{n,k}'$ such that $KG_{n,k}'$ and the Kneser graph $KG_{n,k}$ are not isomorphic but their Kronecker double coverings are isomorphic.
Kronecker classes, normal coverings and chief factors of groups
本文修正了普雷杰关于有限群A-覆盖子群的定理证明，探讨了克罗内克类与群结构的关系。
New completely regular q-ary codes based on Kronecker products
对任意整数$ρ\geq 1$和任意素数幂q，给出了一个覆盖半径为$ρ$，d=3的完全正则（且完全传递）q元码的无限族的显式构造.还计算交集数组。在同样的条件下，给出了覆盖半径为$ρ$的非完全正则的q元均匀填充码（广义）的显式构造.在这两种构造中，克罗内克积都是使用的基本工具。
The family of generalized Petersen graphs $G(n, k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $\mathrm{KC}(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe tha…
Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph
研究了n阶完全图的两列整矩阵A和关联矩阵的Kronecker积的所有子行列式的最小公倍数lcmd（A × B）.我们证明了这个量是lcmd（A）的n-1次方的最小公倍数和A的某些项的二项式函数。
Kronecker Products, Polarity Quotients and Large Graph Constructions
本文建立带极性二部图克罗内克积与极性商的结构兼容性，应用该理论以广义多边形为因子构造出三类大图，其中直径3的低度数图大小超现有同类图。
Further results on multiple coverings of the farthest-off points
研究了射影空间$PG(N,q)$中最远点的多重覆盖（$(R,μ)$-MCF码）和相应的$(ρ,μ)$ -饱和集.我们提出并发展了一些方法，使我们能够获得新的小$(1,μ)$饱和集和短$(2,μ)$码的$μ$密度等于1（最佳饱和集和几乎完美的MCF码）或接近1（大致$1+1/cq$，$c\ge1$）。特别是，我们提供了新的代数结构和一些界限。此外，我们分类的最小和最佳的$(1,μ)$饱和集在$PG(2,q)$，$q$小。
Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes.
Multiple coverings with closed polygons
一个平面集合$P$被称为覆盖可分解的，如果存在一个常数$k=k(P)$，使得平面的每一个具有$P$平移的$k$ -折叠覆盖都可以分解为两个覆盖.已知开凸多边形是覆盖可分解的。在这里，我们证明了封闭的，中心对称的凸多边形也是覆盖可分解的。我们还表明，一个无限倍覆盖的平面与$P$的平移可以分解成两个无限倍覆盖。这两个结果保持覆盖的任何子集的平面。
Multiple vertex coverings by specified induced subgraphs
给定图H_1，...，H_k，我们研究了图G的最小阶，使得对每一个i，H_i在G中的诱导副本覆盖V（G）。本文证明了一个一般的上界，其中m_i比H_i的阶数小小于1。当k=2且一个图是一个大小为n的独立集时，我们在一个常数范围内确定最优解。当k=2且图是一个星和一个独立集时，我们精确地确定了答案。
We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series. Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the covering is a two-periodic configuration. If a non-trivial periodizer has line polynomial factors in at most one direction,…
New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code
本研究以三元完美Golay码及其扩展、删截码为起始码，通过m提升构造出5个新的广义均匀填充近MDS码无限族，它们同时是最远点的几乎完美多重覆盖，并证明了相关通用定理。
An Iwasawa-type asymptotic formula for multiple $\mathbb{Z}_p$-coverings of graphs
对于连通图的可能分歧的 $\\\mathbb{Z}_p^d$-覆盖，建立了复杂度 $p$-进赋值的 Iwasawa 型渐近公式，消除了先前工作中的误差项，并给出了 $\\lambda$ 和 $\\mu$ 不变量在 $p$-覆盖下的 Kida 型公式，以及构造了具有指定不变量的未分歧覆盖。
Octants are Cover-Decomposable into Many Coverings
我们证明了八分区是覆盖分解成多个覆盖，即，对于任何k，存在一个m（k），使得空间的任何子集的任何m（k）-重覆盖具有给定八分圆的有限数量的平移，可以分解为k个覆盖。作为推论，我们得到了平面的任何子集的m（k）-重覆盖都可以分解为k个覆盖。以前只有一些较弱的边界已知的相关问题。
Improved Lower Bounds on the Domination Number of Hypercubes and Binary Codes with Covering Radius One
本文针对超立方体支配数问题，利用同余性质提出新方法，在n为6的倍数时将下界从2^n/n改进为(n-2)2^n/(n^2-2n-2)。
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem …
List covering of regular multigraphs with semi-edges
本文研究含半边正则多重图的列表覆盖复杂性，证明价大于2且含半简单顶点时 List-H-Cover 为NP完全，并给出三次图的NP-co/多项式时间二分法。
Voronoi tilings hidden in crystals - The case of maximal abelian coverings
考虑一个有限连通图，它可能有重边和环.在离散几何分析中，Kotani和Sunada构造了与图相关的晶体，作为图的最大阿贝尔覆盖的标准实现。作为作者在早期的一篇论文中所展示的内容的应用，Seshadri作为几何不变理论的副产品，他表明Voronoi平铺（也称为Wigner-Seitz平铺）隐藏在晶体中，也就是说，晶体不会侵入顶维Voronoi单元的内部。其结果与紧热带曲线的热带Abel-Jacobi图密切相关。
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