Graphs with no Induced $K_{2, t}$

Freddie Illingworth

doi:10.37236/9223

Abstract

Consider a graph $G$ on $n$ vertices with $\alpha \binom{n}{2}$ edges which does not contain an induced $K_{2, t}$ ($t \geqslant 2$). How large must $\alpha$ be to ensure that $G$ contains, say, a large clique or some fixed subgraph $H$? We give results for two regimes: for $\alpha$ bounded away from zero and for $\alpha = o(1)$. Our results for $\alpha = o(1)$ are strongly related to the Induced Turán numbers which were recently introduced by Loh, Tait, Timmons and Zhou. For $\alpha$ bounded away from zero, our results can be seen as a generalisation of a result of Gyárfás, Hubenko and Solymosi and more recently Holmsen (whose argument inspired ours).

Highlights

Fix an integer t and consider a α n 2 edges which does not contain an induced K2,t
How large does α have to be to ensure that G contains some substructure? We consider two regimes: α is bounded away from zero and α goes to zero as n goes to infinity
Hubenko and Solymosi [7] dealt with the clique number in the case when t = 2, confirming a conjecture of Erdos

Summary

Introduction

Fix an integer t and consider a α n 2 edges which does not contain an induced K2,t. How large does α have to be to ensure that G contains some substructure (like a large clique or a fixed subgraph H)? We consider two regimes: α is bounded away from zero and α goes to zero as n goes to infinity. Note that β2(α) = 1 − 1 − α so Proposition 2 can be stated as: if G is a graph on n vertices with α n 2 edges containing no induced K2,2, ω(G). Theorem graph on n vertices n 2 edges containing no induced K2,t and let β = βt(α). Consider a graph G on n vertices with α n 2 edges containing no induced K2,t – how large must α be to ensure that some fixed graph H is a subgraph of G? We give a slightly more general result (expressing the upper bound for the induced Turan number in terms of a Ramsey number involving H – see Corollary 15 and Theorem 18) followed by an improvement to the general upper bound. We focus on the regime where α goes to zero and consider the induced Turan numbers introduced by Loh, Tait, Timmons and Zhou. If G does not contain H ω(G) < vH so, by Theorem 7, vH >

Proof of main result

Improvement when there are few triangles