Splits with forbidden subgraphs

Abstract

In this paper, we fix a graph H and ask into how many vertices 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 set is a pairwise disjoint union of n parts of size at most k, such that there is an edge between any two distinct parts. Letf(n,H)=min⁡{k∈N:there is an (n,k)-graph G such that H⊈G}. Barbanera and Ueckerdt [4] 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., ex(n,H)=Θ(nr) for some r, we show that Ω(n2/r−1)=f(n,H)=O(n2/r−1log1/r⁡n). We extend this result to all bipartite graphs for which upper and a lower Turán exponents do not differ by much. In addition, we prove that f(n,K2,t)=Θ(n1/3) for any fixed integer t≥2.