2024/07/02 by Fábio Botler, Botler, Fábio, Tássio Naia +1 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2407.02102
openalex publication_date 2024/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
A separating system of a graph G is a family S of subgraphs of G for which the following holds: for all distinct edges e and f of G, there exists an element in S that contains e but not f. Recently, it has been shown that every graph of order n admits a separating system consisting of 19n paths [Bonamy, Botler, Dross, Naia, Skokan, Separating the Edges of a Graph by a Linear Number of Paths, Adv. Comb., October 2023], improving the previous almost linear bound of O(nlog^⋆ n) [S. Letzter, Separating paths systems of almost linear size, Trans. Amer. Math. Soc., to appear], and settling conjectures posed by Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi, Letzter, and Narayanan. We investigate a natural generalization of these results to subdivisions of cliques, showing that every graph admits both a separating system consisting of 41n edges and cycles, and a separating system consisting of 82 n edges and subdivisions of K4.