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Induced subdivisions with pinned branch vertices

Sepehr Hajebi

Abstract

We prove that for all $r\in \mathbb{N}\cup \{0\}$ and $s,t\in \mathbb{N}$, there exists $Ω=Ω(r,s,t)\in \mathbb{N}$ with the following property. Let $G$ be a graph and let $H$ be a subgraph of $G$ isomorphic to a $(\leq r)$-subdivision of $K_Ω$. Then either $G$ contains $K_t$ or $K_{t,t}$ as an induced subgraph, or there is an induced subgraph $J$ of $G$ isomorphic to a proper $(\leq r)$-subdivision of $K_s$ such that every branch vertex of $J$ is a branch vertex of $H$. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.

Induced subdivisions with pinned branch vertices

Abstract

We prove that for all and , there exists with the following property. Let be a graph and let be a subgraph of isomorphic to a -subdivision of . Then either contains or as an induced subgraph, or there is an induced subgraph of isomorphic to a proper -subdivision of such that every branch vertex of is a branch vertex of . This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.
Paper Structure (3 sections, 8 theorems, 6 equations)

This paper contains 3 sections, 8 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainshort']}
  3. Acknowledgement

Key Result

Theorem 1.1

For all positive integers $r$ and $p$, there is a positive integer $m = m(r,p)$ such that every graph $G$ containing a $(\leq p)$-subdivision of $K_m$ as a subgraph contains either $K_{p,p}$ as a subgraph or a proper $(\leq p)$-subdivision of $K_{r,r}$ as an induced subgraph.

Theorems & Definitions (13)

  • Theorem 1.1: Lozin and Razgon, Theorem 3 in lozin
  • Theorem 1.3
  • Theorem 2.1: Ramsey multiramsey
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • ...and 3 more