Transversal designs and induced decompositions of graphs

Abstract

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as $\left(\genfrac{}{}{0ex}{}{n}{2}\right)-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $\Omega (n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-decomposition.