Induced subdivisions in graphs of large girth

TL;DR

This paper proves that graphs with large minimum degree and girth contain induced subdivisions of complete graphs, answering a longstanding open problem in graph theory.

Contribution

It establishes the existence of an absolute girth bound ensuring induced subdivisions of complete graphs in graphs with large minimum degree.

Findings

Graphs with large girth and minimum degree contain induced subdivisions of K_{k+1}

Introduces an induced variant of Mader's theorem for graphs with large girth

Provides bounds on girth for induced subdivisions in graphs

Abstract

In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta(G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This answers, in a strong sense, a problem asked by K\"uhn and Osthus (originally attributed to Shi). A main ingredient in our proof is an induced variant of Mader's theorem: for every fixed \(s,\eta,D\), every graph \(J\) with \(\Delta(J)\le D\), \(d(J)>s-2+\eta\) and sufficiently large girth contains an induced subdivision of \(K_s\).