A proof of Mader's conjecture on large clique subdivisions in $C_4$-free graphs

TL;DR

This paper proves a conjecture by Mader from 1999, showing that $C_4$-free graphs with high average degree contain large clique subdivisions, advancing understanding of extremal graph theory.

Contribution

It establishes a lower bound on the size of clique subdivisions in $K_{s,t}$-free graphs, confirming Mader's conjecture for $s=2$ and providing tight bounds up to a constant.

Findings

Confirmed Mader's conjecture for $s=2$ in $C_4$-free graphs.

Provided bounds on clique subdivision sizes in $K_{s,t}$-free graphs.

Showed the result is asymptotically tight up to a constant factor.

Abstract

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$.