# Minimal H-factors and covers

## TL;DR

This paper investigates the minimum total weight of H-factors in complete graphs with random edge weights, establishing concentration results based on the structure of H, especially when H contains a cycle.

## Contribution

It provides new sharp concentration results for the minimum weight of H-factors in weighted complete graphs, extending previous work from simple cases like K_2 to more complex graphs H.

## Key findings

- Minimum weight concentrates around L_n = Θ(n^{1-1/d^*}) for graphs H with cycles.
- Results extend to H-covers where H copies are not vertex-disjoint.
- The concentration depends on the maximum 1-density of subgraphs of H.

## 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 $L_n = \Theta(n^{1-1/d^*})$ (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.