Separating path systems in complete graphs

TL;DR

This paper proves the existence of small collections of paths in complete and regular graphs that can distinguish any pair of edges, advancing understanding of path systems in graph theory.

Contribution

It introduces optimal bounds for path collections that strongly separate edge pairs in complete and regular graphs, improving previous results.

Findings

In complete graphs, about n paths suffice for strong edge separation.

For certain regular graphs, approximately (ab  1 + o(1))n paths are enough.

Results are proven to be asymptotically optimal.

Abstract

We prove that in any $n$-vertex complete graph there is a collection $\mathcal{P}$ of $(1 + o(1))n$ paths that strongly separates any pair of distinct edges $e, f$, meaning that there is a path in $\mathcal{P}$ which contains $e$ but not $f$. Furthermore, for certain classes of $n$-vertex $\alpha n$-regular graphs we find a collection of $(\sqrt{3 \alpha + 1} - 1 + o(1))n$ paths that strongly separates any pair of edges. Both results are best-possible up to the $o(1)$ term.