Exact upper bounds for the minimum sizes of strong and weak separating path systems of cliques

TL;DR

This paper establishes tighter upper bounds for the minimum sizes of strong and weak separating path systems in complete graphs, improving previous asymptotic bounds.

Contribution

It provides exact upper bounds of n+9 and n+1 for strong and weak separation numbers of K_n, surpassing earlier asymptotic results.

Findings

Strong separation number ≤ n+9

Weak separation number ≤ n+1

Improves previous bounds from asymptotic to exact

Abstract

We prove an upper bound of $n+9$ for the strong separation number of the complete graph $K_n$, and an upper bound of $n+1$ for its weak separation number. This improves on the previous best known bound of $(1+o(1))n$ for both cases.