An improved minimum degree threshold for fractional clique decompositions in partite graphs

Abstract

We prove a Dirac-type condition for an $r$-partite graph to admit a fractional clique decomposition: every sufficiently large $r$-partite graph with $n$ vertices in each part and partite minimum degree at least $(1 - \frac{1}{30r^2}) n$ admits a fractional $K_r$-decomposition, subject to necessary divisibility conditions. This implies the same upper bound on the asymptotic minimum degree threshold for $r$-partite graphs to admit a $K_r$-decomposition, using a well-known reduction.