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

TL;DR

The work addresses separating the edges of a graph by a small family of subgraphs and generalizes this notion to subdivisions of cliques. It proves a $K_4$-separating system of size at most $82n$ for any $n$-vertex graph by reducing to a spanning subdivision of $K_4$ with a Hamilton cycle, partitioning edge sets into linear families $M_k$ and $N_k$, and covering each with $K_4$ subdivisions; it further leverages a $K_4$-cover bound of at most $2n-3$. A cycle-based version is established with a $41n$ bound using Pyber’s cycle/edge covering theorem, and the subdivision-to-cycle relation yields insights for $K_3$ separation as well. The results inaugurate the study of linear-size separating systems built from edges and subdivisions of $K_t$, with promising avenues toward larger cliques and related structures like contractions or immersions.

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 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 $K_4$.

Figures (4)

• Figure 1: A graph consisting of a cycle together with a matching $M \subseteq \{u_iu_j: j-i = 4\}$ (in dashed red) having no cycle that contains all edges of $M$.
• Figure 2: a path (highlighted) obtained by an elementary exchange fixing $v$.
• Figure 3: Left: a set $S$ in a Hamiltonian graph and its neighborhood $N(S)$; part of a Hamiltonian cycle is highlighted, dashed red edges are the edges in $C_S$. Right: cycles and edges that separate subgraphs of $G$, where $A\stackrel{\alpha}{\to}B$ indicates that $\alpha$ separates $A$ from $B$ (for instance, $\mathcal{Q'}$ separates $G'$ from $H'$).
• Figure 7: Thick lines illustrate the path $Q_r'$ that contains $M_r\subseteq N_{6,t,r}$ for some $t$ and $r$.

Theorems & Definitions (8)

• Conjecture 1: balogh2016pathfalgas2013separating
• Theorem 2
• Conjecture 3
• Theorem 4
• Lemma 5: brandt2006global
• Theorem 6: jorgensen1990covering
• proof : Proof of Theorem \ref{['thm:K4-separation']}
• Theorem 7: pyber