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Packing $A$-paths of length zero modulo a prime

Robin Thomas, Youngho Yoo

Abstract

It is known that $A$-paths of length $0$ mod $m$ satisfy the Erdős-Pósa property if $m=2$ or $m=4$, but not if $m > 4$ is composite. We show that if $p$ is prime, then $A$-paths of length $0$ mod $p$ satisfy the Erdős-Pósa property. More generally, in the framework of undirected group-labelled graphs, we characterize the abelian groups $Γ$ and elements $\ell \in Γ$ for which the Erdős-Pósa property holds for $A$-paths of weight $\ell$.

Packing $A$-paths of length zero modulo a prime

Abstract

It is known that -paths of length mod satisfy the Erdős-Pósa property if or , but not if is composite. We show that if is prime, then -paths of length mod satisfy the Erdős-Pósa property. More generally, in the framework of undirected group-labelled graphs, we characterize the abelian groups and elements for which the Erdős-Pósa property holds for -paths of weight .

Paper Structure

This paper contains 12 sections, 30 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $A$-paths of a fixed weight in group-labelled graphs
  3. $A$-paths of a fixed weight in infinite groups
  4. $\Gamma$-zero $A$-paths in directed $\Gamma$-labelled graphs
  5. $\Gamma$-zero $A$-paths in undirected $\Gamma$-labelled graphs
  6. $A$-paths of a fixed weight in undirected group-labelled graphs
  7. Preliminaries
  8. Tangles, $K_m$-models, and walls
  9. $A$-paths of length zero modulo a prime
  10. $\Gamma$-nonzero $A$-cycle-chains
  11. $\Gamma$-Bipartite 3-block
  12. Proof of Theorem \ref{['apathsmodptheorem']}

Key Result

Theorem \oldthetheorem

Let $\Gamma$ be a group and let $(\vec{G},\gamma)$ be a directed $\Gamma$-labelled graph with $A \subseteq V(G)$. Then for all positive integers $k$, either $(\vec{G},\gamma)$ contains $k$ disjoint $\Gamma$-nonzero $A$-paths or there is a set of at most $2k-2$ vertices intersecting every $\Gamma$-no

Figures (4)

  • Figure 1: The black vertices constitute $A$ and all unlabelled edges have weight 0.
  • Figure 2: An elementary 6-wall. The four corners are marked by square vertices and its top nails are the vertices filled black. The third vertical path is marked bold and the fourth horizontal path is highlighted in grey.
  • Figure 3: The black filled vertices are in $B_\ell$ and the highlighted curves represent edges of $B_\ell$. If $a$ is not adjacent to another $V(B)$-bridge of $G-A$ attaching to $b_1$, then an $A$-$U$-path in $B_\ell$ starting with the vertices $a,b_1,b_2$ does not have a corresponding path of weight $\ell$ in $(G,\gamma)$, and is improper.
  • Figure 6: The highlighted path is an $A$-$N_1$-path of weight $\ell$ in $(H-Y,\gamma)$ as described in the proof of Claim \ref{['claim:AN1path']}. At least one such path is disjoint from $Y$.

Theorems & Definitions (56)

  • Theorem \oldthetheorem: Theorem 1.1 in ChuGeeGer
  • Theorem \oldthetheorem: Theorem 1 in BruUlm
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 46 more