Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions

TL;DR

This paper proves that dense graphs with high minimum degree can be nearly perfectly packed with complete subgraphs of size k, using fractional decompositions and packings, approaching optimality asymptotically.

Contribution

It establishes asymptotically optimal $K_k$-packings in dense graphs via fractional $K_k$-decompositions, extending understanding of graph packing in dense regimes.

Findings

Graphs with high minimum degree admit fractional $K_k$-decompositions.

Such graphs have $K_k$-packings covering all but $o(n^2)$ edges.

Results hold for all fixed $k > 2$ with explicit degree bounds.

Abstract

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-packing} of a graph $G$ is a set of edge disjoint copies of $H$ in $G$. The following results are proved. For every fixed $k > 2$, every graph with $n$ vertices and minimum degree at least $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges.