A $k$-matching $M$ of a graph $G=(V,E)$ is a subset $M\subseteq E$ such that each connected component in the subgraph $F = (V,M)$ of $G$ is either a single-vertex graph or $k$-regular, i.e., each vertex has degree $k$. In this contribution, we are interested in $k$-matchings within the four standard graph products: the Cartesian, strong, direct and lexicographic product. As we shall see, the problem of finding non-empty $k$-matchings ($k\geq 3$) in graph products is NP-complete. Due to the general intractability of this problem, we focus on distinct polynomial-time constructions of $k$-matchings in a graph product $G\star H$ that are based on $k_G$-matchings $M_G$ and $k_H$-matchings $M_H$ of its factors $G$ and $H$, respectively. In particular, we are interested in properties of the factors that have to be satisfied such that these constructions yield a maximum $k$-matching in the respective products. Such constructions are also called "well-behaved" and we provide several characterizations for this type of $k$-matchings. Our specific constructions of $k$-matchings in graph products satisfy the property of being weak-homomorphism preserving, i.e., constructed matched edges in the product are never "projected" to unmatched edges in the factors. This leads to the concept of weak-homomorphism preserving $k$-matchings. Although the specific $k$-matchings constructed here are not always maximum $k$-matchings of the products, they have always maximum size among all weak-homomorphism preserving $k$-matchings. Not all weak-homomorphism preserving $k$-matchings, however, can be constructed in our manner. We will, therefore, determine the size of maximum-sized elements among all weak-homomorphims preserving $k$-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.

中文翻译：

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图产品中 $k$-matchings 和 $k$-regular 子图的构建

图 $G=(V,E)$ 的 $k$-匹配 $M$ 是一个子集 $M\subseteq E$，使得子图中的每个连通分量 $F = (V,M)$ 的 $G $ 要么是单顶点图，要么是 $k$-regular，即每个顶点都有度数 $k$。在这个贡献中，我们对四种标准图形产品中的 $k$-匹配感兴趣：笛卡尔、强、直接和词典产品。正如我们将看到的，在图产品中找到非空的 $k$-matchings ($k\geq 3$) 的问题是 NP 完全的。由于这个问题的普遍性，我们专注于基于 $k_G$-matchings $M_G$ 和 $k_H$- 的图产品 $G\star H$ 中 $k$-matchings 的不同多项式时间构造分别匹配其因子 $G$ 和 $H$ 的 $M_H$。特别是，我们对必须满足的因素的属性感兴趣，以便这些构造在各自的产品中产生最大的 $k$ 匹配。这种结构也被称为“行为良好”，我们为这种类型的 $k$ 匹配提供了几个特征。我们在图乘积中 $k$-matchings 的特定构造满足弱同态保留的特性，即乘积中构造的匹配边永远不会“投影”到因子中的不匹配边。这导致了弱同态保留 $k$-matchings 的概念。虽然这里构建的特定 $k$-匹配并不总是产品的最大 $k$-匹配，但它们总是在所有弱同态保留 $k$-匹配中具有最大尺寸。然而，并非所有弱同态都保留 $k$-匹配，可以按照我们的方式构建。因此，如果因子中的匹配满足一些一般假设，我们将确定所有弱同态中最大尺寸元素的大小，并在各自的图产品中保留 $k$-matching。