On the Tur\'an number of the blow-up of the hexagon

TL;DR

This paper investigates the Turán number of the blow-up of certain bipartite graphs, establishing tight bounds for specific cases like the hexagon and theta graphs, advancing understanding in extremal graph theory.

Contribution

It proves tight bounds on the Turán number of the 2-blowup of the hexagon and theta graphs, a significant step in bipartite graph extremal problems.

Findings

Proves that ex(n,C6[2])=O(n^{5/3})

Establishes bounds for ex(n,θ_{3,t}[2])=O(n^{5/3}) for large t

Results are tight for sufficiently large t

Abstract

The $r$-blowup of a graph $F$, denoted by $F[r]$, is the graph obtained by replacing the vertices and edges of $F$ with independent sets of size $r$ and copies of $K_{r,r}$, respectively. For bipartite graphs $F$, very little is known about the order of magnitude of the Tur\'an number of $F[r]$. In this paper we prove that $\mathrm{ex}(n,C_6[2])=O(n^{5/3})$ and, more generally, for any positive integer $t$, $\mathrm{ex}(n,\theta_{3,t}[2])=O(n^{5/3})$. This is tight when $t$ is sufficiently large.