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Induced subdivisions of $K_{d+1}$ in graphs of high girth

António Girão, Zach Hunter

Abstract

In this paper, we show that for all $k\geq 10^8$, every graph with minimum degree $k$ and girth at least $10^8$ contains an induced subdivision of a $K_{k+1}$. This answers a problem asked by Kühn and Osthus (originally attributed to Shi).

Induced subdivisions of $K_{d+1}$ in graphs of high girth

Abstract

In this paper, we show that for all , every graph with minimum degree and girth at least contains an induced subdivision of a . This answers a problem asked by Kühn and Osthus (originally attributed to Shi).
Paper Structure (7 sections, 11 theorems, 4 equations)

This paper contains 7 sections, 11 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main proofs
  4. Key proposition
  5. Proof of theorem
  6. Case 1: |A'| >= n/2
  7. Case 2: |A'| <= n/2

Key Result

Theorem 1.1

Let $G$ be a graph with minimum degree $d\geqslant 10^8$ and girth at least $10^8$. Then $G$ contains an induced subdivision of $K_{d+1}$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1: Lovász Local Lemma ErdosLovasz1975LLLAlonSpencerProbMethod
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 15 more