Fractional Clique Decompositions of Dense Graphs and Hypergraphs

TL;DR

This paper proves that dense graphs and hypergraphs with high minimum degree conditions admit fractional decompositions into cliques, providing new combinatorial proofs and improving thresholds for exact decompositions.

Contribution

It establishes new minimum degree thresholds for fractional clique decompositions in dense graphs and hypergraphs, and offers a purely combinatorial proof of Wilson's theorem.

Findings

Dense graphs with high minimum degree have fractional clique decompositions.

Hypergraphs with high minimum codegree admit fractional clique decompositions.

Provides combinatorial proofs for classical decomposition theorems.

Abstract

Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) $F$-decompositions for a wide class of graphs~$F$ (including large cliques). For general $k$-uniform hypergraphs, we give a short argument which shows that there exists a constant $c_k>0$ such that every $k$-uniform hypergraph $G$ on $n$ vertices with minimum codegree at least $(1- c_k /r^{2k-1}) n $ has a fractional $K^{(k)}_r$-decomposition, where $K^{(k)}_r$ is the complete $k$-uniform hypergraph on $r$ vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large $F$-divisible complete graph has an $F$-decomposition.