A coarse Gallai theorem

TL;DR

This work establishes a coarse analogue of Gallai’s path-packing theorem: for any $k,d$, in every graph $G$ and subset $A$, either one can find $k$ $A$-paths pairwise at distance at least $d$, or there exists a hitting set $X$ of size at most $f(k)$ such that every $A$-path lies within distance $g(d,k)$ of $X$. The authors develop a frame-based, coarse geometric approach built around fat models of subcubic forests and a central Tripod Lemma to extend frames, yielding explicit bounds $f(k)=4k-4$ and $g(k,d)=256^{k}d$. They also prove a coarse version of the minor-to-topological-minor correspondence and discuss algorithmic aspects and conjectures about radius bounds depending only on $d$, connecting Gallai-type packing to coarse Menger/Erdős–Pósa themes. Overall, the paper advances coarse graph theory by providing constructive packing/hitting dichotomies with concrete constants and a robust framework to translate minor-theoretic ideas into large-scale graph structure.

Abstract

We prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths such that vertices of different $A$-paths are at distance at least $d$ in $G$, or there exists a set $X$ of the vertices of $G$ with $|X|\leq f(k)$ such that every $A$-path in $G$ contains a vertex of $B_G(X,g(k,d))$.

Figures (13)

• Figure 1: The \ref{['lemma:tripod']}. The vertices $v_1$, $v_2$, $v_3$ are not too close and not too far from $V(Q)$, and they are far from each other. The subgraphs $Z$, $P_1$, $P_2$, $P_3$ of $G$ are the outcome of the lemma.
• Figure 2: A tripoid $(C,\xi,\{R_i,w_i,B_i\}_{i\in[3]})$.
• Figure 3: Construction of $Z$, $P_1$, $P_2$, $P_3$ when $\mathrm{dist}_G(V(R_\alpha), V(B_\xi)) < \ell$.
• Figure 4: Construction of $Z$, $P_1$, $P_2$, $P_3$ when $\mathrm{dist}_G(V(B_\alpha), V(B_\beta)) < \ell$.
• Figure 5: Construction of $(D, \alpha,\{(R_i',w_i',B_i')\}_{i\in[3]})$ within the proof of the \ref{['lemma:tripod']}. On the left, we depict the case where $\mathrm{dist}_G(w_\alpha,V(D)) = \ell$ and on the right, we depict the case where $\mathrm{dist}_G(w_\alpha,V(D)) > \ell$. We highlight in blue the new path $B_\alpha'$ and in red the new path $R_\alpha'$. In the depicted cases, $c_\alpha$ is a cut-vertex of $C$. The light green bubbles are components of $C - c_\alpha$, one of them is $D$.

Theorems & Definitions (23)

• theorem 1
• theorem 2
• theorem 3
• Lemma 4: Tripod Lemma
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7