Minimal H-factors and covers
Lorenzo Federico, Joel Larsson Danielsson
TL;DR
This paper analyzes the minimum-weight tiling of a fixed graph $H$ into a complete graph $K_n$ with i.i.d. edge weights, focusing on $H$-factors (vertex-disjoint copies covering all vertices) and $H$-covers (overlapping copies allowed). The authors develop a divide-and-conquer framework using a red-green edge-splitting trick to construct large low-cost partial $H$-factors and complete them with small additional cost, enabling sharp concentration around a scale $L_n=\Theta(n^{1-1/d^*})$, where $d^* = \max_{G\subseteq H} d_G$ is the maximum $1$-density of a subgraph of $H$. They prove lower and upper bounds, establish a concentration result around a median $M$ with $|F_H(\alpha n,n)-M|=O_{\mathbb{P}}(M^{3/4})$, and extend the discussion to $H$-covers, noting that sharp concentration may fail in unbalanced cases. Furthermore, they show the results hold under general edge-weight distributions with pseudo-dimension 1, via a coupling argument, and provide a detailed treatment of when $H$ is not balanced. The work advances understanding of random-weight tiling problems and demonstrates robust concentration phenomena for a broad class of structured graph completions.
Abstract
Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, 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 $L_n = Θ(n^{1-1/d^*})$ (where $d^*$ 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.