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

01/20/2023

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by   Marthe Bonamy, et al.

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Recently, Letzter proved that any graph of order n contains a collection 𝒫 of O(nlog^⋆ n) paths with the following property: for all distinct edges e and f there exists a path in 𝒫 which contains e but not f. We improve this upper bound to 19 n, thus answering a question of 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.