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Edge mappings of graphs: Ramsey type parameters

Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer

Abstract

In this paper, we address problems related to parameters concerning edge mappings of graphs. Inspired by Ramsey's Theorem, the quantity $m(G, H)$ is defined to be the minimum number $n$ such that for every $f: E(K_n) \rightarrow E(K_n)$ either there is a fixed copy of $G$ with $f ( e) = e$ for all $e\in E(G)$, or a free copy of $H$ with $f( e) \notin E(H)$ for all $e\in E(H)$. We extend many old results from the 80's as well as proving many new results. We also consider several new interesting parameters with the same spirit.

Edge mappings of graphs: Ramsey type parameters

Abstract

In this paper, we address problems related to parameters concerning edge mappings of graphs. Inspired by Ramsey's Theorem, the quantity is defined to be the minimum number such that for every either there is a fixed copy of with for all , or a free copy of with for all . We extend many old results from the 80's as well as proving many new results. We also consider several new interesting parameters with the same spirit.
Paper Structure (14 sections, 33 theorems, 36 equations)

This paper contains 14 sections, 33 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Parameters related to Ramsey numbers.
  4. Our results.
  5. $g(G,1)$ and $g(G,0)$.
  6. w(G)
  7. Organization of the paper
  8. Preliminaries and general connections between parameters
  9. The appearance of $f$-fixed copies of $G$ or $f$-free/exclusive copies of $H$: $m(G,H)$ and $m^*(G,H)$
  10. Results about $m(G,K_r)$
  11. Results about $m(G,K_{1,r})$
  12. The appearance of $f$-free copies of $G$: $g(G)$
  13. The appearance of $f$-exclusive copies of $G$: $w(G)$
  14. Closing remarks

Key Result

Theorem 1

Theorems & Definitions (57)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 47 more