Minimal H-factors and covers

Abstract

Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H^{\prime }\subset G$, each isomorphic to H, that cover the vertices of G. If G is the complete graph $K_n$ 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 e.g.\ $H=K_2$. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n=\Theta (n^{1-1/d^{\ast }})$ (where $d^{\ast }$ 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.