Separating the edges of a graph by cycles and by subdivisions of $K_4$

TL;DR

This paper studies how to separate edges of a graph using specific subgraphs like paths, cycles, and subdivisions of $K_4$, providing improved bounds and generalizations of previous results.

Contribution

It extends the concept of separating systems to subdivisions of cliques, establishing new bounds for separating edges with cycles and subdivisions of $K_4$.

Findings

Every graph admits a separating system of 41n edges and cycles.

Every graph admits a separating system of 82n edges and subdivisions of $K_4$.

Improved bounds settle previous conjectures.

Abstract

A separating system of a graph $G$ is a family $\mathcal{S}$ of subgraphs of $G$ for which the following holds: for all distinct edges $e$ and $f$ of $G$, there exists an element in $\mathcal{S}$ 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 $\mathrm{O}(n\log^\star n)$ [S. Letzter, Separating paths systems of almost linear size, Trans. Amer. Math. Soc., to appear], and settling conjectures posed by Balogh, Csaba, Martin, and Pluh\'ar and by Falgas-Ravry, Kittipassorn, Kor\'andi, 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 $K_4$.