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Graphs of large girth

Christian Reiher

Abstract

This survey on graphs of large girth consists of two parts. The first deals with some aspects of algebraic and extremal graph theory loosely related to the Moore bound. Our point of departure for the second, Ramsey theoretic, part are some constructions of graphs with large chromatic number and large girth; this will lead us to a discussion of the recent girth Ramsey theorem. Both parts can be enjoyed independently of each other.

Graphs of large girth

Abstract

This survey on graphs of large girth consists of two parts. The first deals with some aspects of algebraic and extremal graph theory loosely related to the Moore bound. Our point of departure for the second, Ramsey theoretic, part are some constructions of graphs with large chromatic number and large girth; this will lead us to a discussion of the recent girth Ramsey theorem. Both parts can be enjoyed independently of each other.
Paper Structure (13 sections, 25 theorems, 56 equations, 15 figures)

This paper contains 13 sections, 25 theorems, 56 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Girth, degrees, and the number of vertices
  3. Moore graphs
  4. Cages and upper bounds
  5. Average degree
  6. Directed graphs
  7. The chromatic number
  8. A theorem of Erdős
  9. Historical constructions
  10. The partite construction method
  11. A conjecture of Erdős
  12. Vertex colourings and Ramsey theory
  13. Infinite graphs

Key Result

Theorem 1.1

For every graph $F$ that is not a forest and every number of colours $r$ there exists a graph $H$ of the same girth as $F$ such that for every $r$-colouring of $H$ there is a monochromatic induced copy of $F$.

Figures (15)

  • Figure 2.1: Proof of the Moore bound
  • Figure 2.2: Edge, Pentagon, and Petersen graph
  • Figure 2.3: The smallest projective plane and a tiling of the torus (black rhombus whose opposite sides are identified) with seven hexagons
  • Figure 2.4: Desargues's theorem states that the three points on the dashed line are collinear, provided that the nine triples on the solid lines are.
  • Figure 2.5: Proof of the Erdős-Sachs theorem.
  • ...and 10 more figures

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 2.1: Moore bound
  • proof
  • Theorem 2.2: Hoffman & Singleton
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Conjecture 2.6: Strong prime power conjecture
  • Conjecture 2.7: Weak prime power conjecture
  • ...and 45 more