Minimal H-factors and covers

Journal article, 2025

Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs, each isomorphic to H, that cover the vertices of G. If G is the complete graph equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for, for example. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some (where is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint.