Separating Path Systems for the Complete Graph

TL;DR

This paper studies the minimal size of path collections in complete graphs that can distinguish every pair of edges, providing new bounds and constructions, especially for prime and small cases.

Contribution

It introduces the concept of Generator Paths to construct small separating path systems and establishes bounds for all large and specific small complete graphs.

Findings

Constructed separating path systems with size at most (21/16+o(1))n for large n.

Proved that for prime n, the minimal size is at most n.

Established that for n=p or p+1 with p prime, the size is at most n.

Abstract

For any graph $G$, a separating path system of $G$ is a family of paths in $G$ with the property that for any pair of edges in $E(G)$ there is at least one path in the family that contains one edge but not the other. We investigate the size of the smallest separating path system for $K_n$, denoted $f(K_n)$. Our first main result is a construction that shows $f(K_n) \leq \left(\frac{21}{16}+o(1)\right)n$ for sufficiently large $n$. We also show that $f(K_n) \leq n$ whenever $n=p,p+1$ for prime $p$. It is known by simple argument that $f(K_n) \geq n-1$ for all $n \in \mathbb{N}$. A key idea in our construction is to reduce the problem to finding a single path with some particular properties we call a Generator Path. These are defined in such a way that the $n$ cyclic rotations of a generator path provide a separating path system for $K_n$. Hence existence of a generator path for some $K_n$ gives $f(K_n) \leq n$. We construct such paths for all $K_n$ with $n \leq 20$, and show that generator paths exist whenever $n$ is prime.