Separating the edges of a graph by a linear number of paths

TL;DR

This paper improves the upper bound on the number of paths needed to separate edges in any graph from a logarithmic factor to a linear function of the number of vertices, providing a simpler proof and confirming a longstanding conjecture.

Contribution

The authors establish a new linear upper bound of 19n paths for separating edges in any graph, improving previous bounds and confirming a conjecture.

Findings

Upper bound improved to 19n paths

Answer to a longstanding open question

Elementary and self-contained proof

Abstract

Recently, Letzter proved that any graph of order $n$ contains a collection $\mathcal{P}$ of $O(n\log^\star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $\mathcal{P}$ which contains $e$ but not $f$. We improve this upper bound to $19 n$, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluh\'ar and by Falgas-Ravry, Kittipassorn, Kor\'andi, Letzter, and Narayanan. Our proof is elementary and self-contained.