Splits with forbidden subgraphs

TL;DR

This paper investigates the maximum size of parts in split graphs avoiding a fixed subgraph H, providing bounds based on Turán exponents for bipartite graphs and exact results for complete bipartite graphs.

Contribution

It establishes bounds on the splitting size function f(n,H) for bipartite graphs using Turán exponents and extends results to graphs with similar Turán properties, including exact bounds for K_{2,t}.

Findings

f(n,H) is constant 2 for non-bipartite H.

Bounds on f(n,H) based on Turán exponents for bipartite H.

Exact order of f(n,K_{2,t}) as Θ(n^{1/3}).

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 Tur\'an exponent, i.e., ${\rm ex}(n, H) = \Theta(n^r)$ for some $r$, we show that $\Omega (n^{2/r -1}) = f(n, H) = O (n^{2/r-1} \log ^{1/r} n)$. We extend this result to all bipartite graphs for which an upper and a lower Tur\'an exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =\Theta(n^{1/3})$ for any fixed $t$.