A coarse Gallai theorem

TL;DR

This paper establishes a generalized Gallai theorem demonstrating that in any graph, either many well-separated A-paths exist or a small vertex set intersects all such paths, with bounds depending on parameters.

Contribution

It introduces functions f and g that guarantee either the existence of multiple distant A-paths or a small vertex set intersecting all A-paths, extending Gallai's theorem.

Findings

Existence of functions f and g with specified properties

Either many distant A-paths or a small vertex set intersecting all A-paths

Generalization of Gallai's theorem to broader graph settings

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 $B_G(X,g(k,d))$.