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

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 $19n$, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi, Letzter, and Narayanan. Our proof is elementary and self-contained.

7 pages, 3 figures

References in corpus (2)

• Separating paths systems of almost linear size
• Towards the Erdős-Gallai Cycle Decomposition Conjecture