Construction of k-matchings in graph products

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