Transversal designs and induced decompositions of graphs

TL;DR

This paper demonstrates that very dense graphs can be decomposed into induced subgraphs isomorphic to any complete multipartite graph, revealing a structural gap in the minimum non-edges needed for such decompositions.

Contribution

It establishes the existence of dense graphs with specific edge counts that admit induced decompositions into complete multipartite graphs, clarifying a previously unknown structural gap.

Findings

Existence of dense graphs with ${nrace 2}-cn$ edges decomposable into induced $F$

Identification of a gap between $O(n)$ and $ ext{} ext{Omega}(n^{3/2})$ in non-edges for $F$-decompositions

Structural explanation of the minimum non-edges in graphs with induced $F$-decompositions

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 ${n\choose 2}-cn$ 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.