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Packing cycles in undirected group-labelled graphs

Robin Thomas, Youngho Yoo

Abstract

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,γ)$ where $γ$ assigns to each edge of an undirected graph $G$ an element of an abelian group $Γ$. As a consequence, we prove that $Γ$-nonzero cycles (cycles whose edges sum to a non-identity element of $Γ$) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that, if $Γ$ has no element of order two, then $Γ$-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if $m$ is an odd prime power, then cycles of length $\ell \mod m$ satisfy the Erdős-Pósa property for all integers $\ell$. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs $(\ell,m)$.

Packing cycles in undirected group-labelled graphs

Abstract

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs where assigns to each edge of an undirected graph an element of an abelian group . As a consequence, we prove that -nonzero cycles (cycles whose edges sum to a non-identity element of ) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that, if has no element of order two, then -nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if is an odd prime power, then cycles of length satisfy the Erdős-Pósa property for all integers . This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs .

Paper Structure

This paper contains 15 sections, 32 theorems, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group-labelled graphs
  4. Tangles
  5. 3-blocks
  6. $K_m$-models
  7. Walls
  8. Flat walls
  9. Linkages
  10. Main theorem
  11. Proof of Theorem \ref{['nzcyclesEPtheorem']}
  12. Cycles of length $\ell$ mod $m$
  13. Proof of Theorem \ref{['flatwallundirectedtheorem']}: large $K_t$-model
  14. Proof of Theorem \ref{['flatwallundirectedtheorem']}: large flat wall
  15. Proof of Theorem \ref{['flatwallundirectedtheorem']}

Key Result

Theorem \oldthetheorem

Let $\Gamma$ be an abelian group. Then the family of $\Gamma$-nonzero cycles satisfies the half-integral Erdős-Pósa property. Moreover, if $\Gamma$ has no element of order two, then the family of $\Gamma$-nonzero cycles satisfies the Erdős-Pósa property.

Figures (7)

  • Figure 1: Since $(C_u,D_u)$ is good, if $|V(C_u\cap D_u)|=2$, then there is a path in $C_u$ connecting the two vertices of $V(C_u\cap D_u)$. In (b), $(C\cup C_u, D\cap D_u)$ is not good, so there is a 1-separation $(C',D')$ with $V(C'\cap D')=\{u'\}$ which violates the tightness of $(C_u,D_u)$.
  • Figure 2: An elementary $6\times 7$-wall. The four corners are marked by square vertices and its top nails are filled black. The fourth horizontal path is highlighted in grey and the third vertical path is marked bold.
  • Figure 3: The three black vertices are $w_1,w_2$, and $w_3$. Since $B_{2,1}$ is a $\Gamma$-nonzero cycle, at least one of these three cycle-chains is $\Gamma$-nonzero.
  • Figure 4: A $b(\mu)$-path $P$ in a $K_5$-model $\mu$. Each ellipse indicates a tree $\mu(v_i)$.
  • Figure 5: The tree $T=Q_1^1 \cup Q_2^1 \cup Q_1^2 \cup Q_2^2 \cup Q$. The dashed lines indicate some possible $V(T)$-subpaths $P$ of $P_1$.
  • ...and 2 more figures

Theorems & Definitions (66)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Theorem 1.1 in WolPath
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 56 more