Fractional Clique Decompositions of Dense Partite Graphs

TL;DR

This paper establishes a minimum degree condition that guarantees the existence of fractional clique decompositions in dense balanced r-partite graphs, extending previous non-partite and specific cases.

Contribution

It generalizes fractional clique decomposition conditions to balanced r-partite graphs, building on prior non-partite and 3-partite results.

Findings

Provides a minimum degree threshold for fractional $K_r$-decomposition

Extends results to balanced r-partite graphs

Connects fractional and non-fractional $K_r$-decompositions

Abstract

We give a minimum degree condition sufficent to ensure the existence of a fractional $K_r$-decomposition in a balanced $r$-partite graph (subject to some further simple necessary conditions). This generalises the non-partite problem studied recently by Barber, Lo, K\"uhn, Osthus and the author, and the $3$-partite fractional $K_3$-decomposition problem studied recently by Dukes. Combining our result with recent work by Barber, K\"uhn, Lo, Osthus and Taylor, this gives a minimum degree condition sufficient to ensure the existence of a (non-fractional) $K_r$-decomposition in a balanced $r$-partite graph (subject to the same simple necessary conditions).