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On the edge-Erdős-Pósa property of Ladders

Raphael Steck, Arthur Ulmer

Abstract

We prove that the ladder with $3$~rungs and the house graph have the edge-Erdős-Pósa property, while ladders with $14$~rungs or more have not. Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit a better result.

On the edge-Erdős-Pósa property of Ladders

Abstract

We prove that the ladder with ~rungs and the house graph have the edge-Erdős-Pósa property, while ladders with ~rungs or more have not. Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit a better result.

Paper Structure

This paper contains 12 sections, 26 theorems, 2 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Long Ladders
  4. Finding one ladder
  5. Excluding two ladders
  6. Proof of Theorem \ref{['no14rungs']}
  7. Small Ladder
  8. Useful Techniques
  9. $(A,m)$-Trees
  10. Graphs without Ladders (or Houses)
  11. Proof of Theorem \ref{['3RungLadderHasEdgeEPProperty']}
  12. Discussion

Key Result

Theorem 1

The ladder with $3$ rungs has the edge-Erdős-Pósa property.

Figures (19)

  • Figure 1: A ladder with 5 rungs
  • Figure 2: A condensed wall of size 5.
  • Figure 3: Construction in Theorem \ref{['no14rungs']} for a ladder with 14 rungs. Note that all edges drawn represent $r$ internally disjoint paths of length $2$.
  • Figure 4: An ($a$-$b$, $c$-$d$)-linkage in an X-wing
  • Figure 5: An X-wing (thick edges) in a condensed wall of size 2. Edges belonging to rungs are labelled and red.
  • ...and 14 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 1
  • Conjecture 2: Bruhn, Heinlein and Joos bruhn18
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Bruhn et al. bruhn18
  • Lemma 6
  • proof
  • ...and 38 more