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Graphs with all holes the same length

Linda Cook, Jake Horsfield, Myriam Preissmann, Cléophée Robin, Paul Seymour, Ni Luh Dewi Sintiari, Nicolas Trotignon, Kristina Vušković

Abstract

A graph is "$\ell$-holed" if all its induced cycles of length at least four have length exactly $\ell$. We give a complete description of the $\ell$-holed graphs for each $\ell\ge 7$.

Graphs with all holes the same length

Abstract

A graph is "-holed" if all its induced cycles of length at least four have length exactly . We give a complete description of the -holed graphs for each .

Paper Structure

This paper contains 12 sections, 42 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The "if" half
  3. Blow-ups of cycles
  4. Frames
  5. Gates
  6. Attachments to a maximal frame
  7. Blowing up a frame
  8. Putting pieces together
  9. Border construction
  10. The case when $\ell$ is odd
  11. A structure theorem for laminar families
  12. The case when $\ell$ is even

Figures (9)

  • Figure 1: A blow-up of a 7-cycle.
  • Figure 2: An 18-bar 9-framework.
  • Figure 3: The first pair are coarboreal, the second pair are not.
  • Figure 4: An 18-bar 8-framework.
  • Figure 5: A theta, a pyramid and a prism (dashed lines mean paths of arbitrary positive length)
  • ...and 4 more figures