Fractional clique decompositions of dense balanced multipartite graphs

This paper concerns fractional $K_s$-decompositions of multipartite graphs. For integers $r\ge s\ge 3$, we consider balanced $r$-partite graphs $G$ on $rn$ vertices. We establish necessary conditions for $G$ to admit a fractional $K_s$-decomposition, extending the notion of $s$-admissibility from the case $r=s$ to $r>s$. Using an association scheme on the edge set of a complete $r$-partite graph, we prove that if $r\ge s+2$ and the partite minimum degree of $G$ is at least $(1-c)n$ with $c\le 1/((s-2)(s+1)(s-1)^4)$, then $G$ has a fractional $K_s$-decomposition. For $r=s+1$, we show that under the condition $c\le 1/(3s^3(s-2)^2)$, every $s$-admissible balanced $(s+1)$-partite graph with partite minimum degree at least $(1-c)n$ admits a fractional $K_s$-decomposition. These results provide new degree thresholds for fractional $K_s$-decompositions of multipartite graphs with more than $s$ parts.