Splits with forbidden subgraphs

Abstract

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise disjoint union of $n$ parts of size at most $k$ each such that there is an edge between any two distinct parts. Let $$ f(n,H) = \min \{k \in \mathbb N : \mbox{there is an $(n,k)$-graph $G$ such that $H\not\subseteq G$}\} . $$ Barbanera and Ueckerdt observed that $f(n,H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Turán exponent, i.e., $\mathrm{ex}(n,H)=\hat{I}˜(n^r)$ for some $r$, we show that $\hat{I}\text{c◯}(n^{2/r-1})=f(n,H)=O(n^{2/r-1}{\log }^{1/r}n)$. We extend this result to all bipartite graphs for which an upper and a lower Turán exponents do not differ by much. In addition, we prove that $f(n,K_{2,t})=\hat{I}˜(n^{1/3})$ for any fixed $t$.

12 pages; In v1 and v2, there were missing floors and ceilings in Theorem 5 that in some cases would lead to an incorrect result. In v3, we correct this minor error