Half-integral Erdős-Pósa property for non-null $S$-$T$ paths

TL;DR

The paper proves that non-null $S$-$T$ paths in finite $\Gamma$-labelled graphs satisfy the half-integral Erdős–Pósa property: for any graph $G$, subsets $S,T$ of vertices, and integer $k$, either there exists a congestion-2 packing of $k$ non-null $S$-$T$ paths or a hitting set of size at most $f(k)$, where $f$ depends only on $\Gamma$. The authors first solve the problem for $(q,k)$-unbreakable graphs (highly connected cases) using a tripod-based construction and a result of Chudnovsky et al., then reduce the general case to the unbreakable one via gadget replacement guided by a new combinatorial notion of graph types that preserve packing/hitting numbers. A finite, albeit non-constructive, notion of type enables shrinking large regions while maintaining relevant properties, leading to a recursive bound $f(k)$ with $f(0)=0$ and $f(k)=\max(4h(k)+2k-2, k-1+2f(k-1))$, where $h(k)$ comes from the type-bounded analysis. The results generalize the half-integral Erdős–Pósa property to non-null $S$-$T$ paths in group-labelled graphs and recover the corollary for odd $S$-$T$ paths, while highlighting non-constructive and group-dependent aspects of the bound and outlining directions for explicit bounds and directed-case extensions.

Abstract

For a group $Γ$, a $Γ$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $Γ$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is non-null if the product of the labels along the path is not the neutral element of $Γ$. We prove that for every finite group $Γ$, non-null $S$-$T$ paths in $Γ$-labelled graphs exhibit the half-integral Erdős-Pósa property. More precisely, there is a function $f$, depending on $Γ$, such that for every $Γ$-labelled graph $G$, subsets of vertices $S$ and $T$, and integer $k$, one of the following objects exists: a family $\cal F$ consisting of $k$ non-null $S$-$T$ paths in $G$ such that every vertex of $G$ participates in at most two paths of $\cal F$; or a set $X$ consisting of at most $f(k)$ vertices that meets every non-null $S$-$T$ path in $G$. This in particular proves that in undirected graphs $S$-$T$ paths of odd length have the half-integral Erdős-Pósa property.

Figures (1)

• Figure 1: A construction showing that odd $S$--$T$ paths do not exhibit the Erdős--Pósa property. The graph is obtained from the $(2n+1)\times (2n+1)$ grid by attaching a degree-$1$ vertex of $S$ to every second vertex of the left side, a degree-$1$ vertex of $T$ to every second vertex of the right side, and a triangle to every edge of the top side. On one hand, every vertex subset $X$ of size smaller than $n$ avoids at least one row containing vertices of $S$ and $T$, as well as two consecutive columns together with the triangle joining them; hence there is an odd $S$--$T$ path not meeting $X$. On the other hand, every odd $S$--$T$ path has to visit the top side of the grid, hence there are no two disjoint odd $S$--$T$ paths. There is, however, a large congestion-$2$ packing of odd $S$--$T$ paths, highlighted through colors. The example is a slightly adapted construction from the work of Bruhn, Henlein, and Joos henning2018frames, which in turn is inspired by the Escher wall of Lovász and Schrijver, see the work of Reed bruce1999mangoes.

Theorems & Definitions (17)

• Theorem 0
• Definition 1
• Theorem 2: chudnovsky2006packing
• Proposition 3
• proof
• Claim 1
• Claim 2
• Lemma 3: Finiteness
• Lemma 3: Compositionality
• Corollary 4