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A $k$-matching $M$ of a graph $G=(V,E)$ is a subset $Msubseteq E$ 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 ($kgeq 3$) 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 $Gstar H$ 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.
In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set ${1,...,2n}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject
In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by
Computing cohesive subgraphs is a central problem in graph theory. While many formulations of cohesive subgraphs lead to NP-hard problems, finding a densest subgraph can be done in polynomial time. As such, the densest subgraph model has emerged as t
Let $n, k, m$ be positive integers with $ngg mgg k$, and let $mathcal{A}$ be the set of graphs $G$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free coloring of $K_n$
Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting poi