Construction of k-matchings in graph products

Abstract

A k-matching M of a graph G = (V,E) is a subset M ⊆ 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 in 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 ≥ 3) in graph products is NP-complete. Due to the general intractability of this problem, we focus on different polynomial-time constructions of k-matchings in a graph product G ⋆ H that are based on kG-matchings MG and kH-matchings MH 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 allweak-homomorphism preserving k-matching within the respective graph products, provided that the matchings in the factors satisfy some general assumptions.