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.

翻译：最近，Letzter证明了任意$n$阶图都包含一个由$O(n\log^\star n)$条路径组成的集合$\mathcal{P}$，且满足以下性质：对于任意两条不同的边$e$和$f$，$\mathcal{P}$中存在一条路径包含$e$但不包含$f$。我们将这一上界改进至$19n$，从而回答了G.O.H. Katona提出的问题，并证实了Balogh、Csaba、Martin和Pluhár以及Falgas-Ravry、Kittipassorn、Korándi、Letzter和Narayanan独立提出的猜想。我们的证明是初等且自包含的。