Fractional clique decompositions of dense graphs

TL;DR

This paper proves that dense graphs with high minimum degree have fractional clique decompositions, improving bounds and extending results to exact decompositions for graphs with chromatic number at least 4.

Contribution

It establishes nearly tight minimum degree conditions for fractional $K_r$-decompositions in dense graphs, and extends to exact $F$-decompositions for graphs with chromatic number at least 4.

Findings

Improved bounds on minimum degree for fractional $K_r$-decomposition.

First bound tight up to constant multiple of $r$.

Extension to exact $F$-decompositions for graphs with $ ext{chromatic number} \\ge 4$.

Abstract

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1/100r)|G|$ has a fractional $K_r$-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional $K_r$-decomposition given by Dukes (for small $r$) and Barber, K\"uhn, Lo, Montgomery and Osthus (for large $r$), giving the first bound that is tight up to the constant multiple of $r$ (seen, for example, by considering Tur\'an graphs). In combination with work by Glock, K\"uhn, Lo, Montgomery and Osthus, this shows that, for any graph $F$ with chromatic number $\chi(F)\ge 4$, and any $\varepsilon>0$, any sufficiently large graph $G$ with minimum degree at least $(1-1/100\chi(F)+\varepsilon)|G|$ has, subject to some further simple necessary divisibility conditions, an (exact) $F$-decomposition.