On the extremal number of subdivisions

TL;DR

This paper investigates the extremal number of bipartite graphs with maximum degree 2, proving a conjecture for the case r=2 and introducing a new dependent random choice technique.

Contribution

It proves a conjecture for bipartite graphs with maximum degree 2, showing tight bounds on extremal numbers and introducing a novel variant of dependent random choice.

Findings

Confirmed the conjecture for r=2 in bipartite graphs.

Established bounds for graphs with no 4-cycle and maximum degree 2.

Introduced a new variant of the dependent random choice technique.

Abstract

One of the cornerstones of extremal graph theory is a result of F\"uredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this is essentially the only situation in which F\"uredi's result can be tight and prove this conjecture for $r = 2$. More precisely, we show that if $H$ is a $C_4$-free bipartite graph with maximum degree $2$ on one side, then there are positive constants $C$ and $\delta$ such that every graph with $n$ vertices and $C n^{3/2 - \delta}$ edges contains a copy of $H$. This answers a question of Erd\H{o}s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.