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

George Kontogeorgiou, Maya Stein

TL;DR

This paper resolves tight additive upper bounds for separating path systems in complete graphs by a constructive prime-decomposition approach and a generating-path framework. It first builds a weakly separating path system of size $n+1$ on $K_n$ by partitioning $n$ into prime blocks and aggregating level-wise constructions, then enhances the system with eight auxiliary paths to achieve strong separation, yielding $ssp(K_n)≤ n+9$. The methods are explicit and modular, leveraging rotations and linear-forest arrangements to ensure separation across all edges. Overall, the results bring the weak and strong separation numbers of complete graphs closer to the conjectured $(1+o(1))n$ scale and provide concrete, implementable constructions.

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.

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

TL;DR

This paper resolves tight additive upper bounds for separating path systems in complete graphs by a constructive prime-decomposition approach and a generating-path framework. It first builds a weakly separating path system of size on by partitioning into prime blocks and aggregating level-wise constructions, then enhances the system with eight auxiliary paths to achieve strong separation, yielding . The methods are explicit and modular, leveraging rotations and linear-forest arrangements to ensure separation across all edges. Overall, the results bring the weak and strong separation numbers of complete graphs closer to the conjectured scale and provide concrete, implementable constructions.

Abstract

We prove an upper bound of for the strong separation number of the complete graph , and an upper bound of for its weak separation number. This improves on the previous best known bound of for both cases.
Paper Structure (4 sections, 2 theorems, 9 equations, 3 figures)

This paper contains 4 sections, 2 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Proof that $wsp(K_n)\leq n+1$
  4. Proof that $ssp(K_n)\leq n+9$

Key Result

Theorem 1

For every natural number $n$, $wsp(K_n)\leq n+1$.

Figures (3)

  • Figure 1: A generating path for $K_{11}$ with $c=1$.
  • Figure 2: For the sake of providing a manageable example, suppose that we decide to needlessly split $K_{14}$ into two levels. Then the present figure depicts a choice for $\mathbf{P_1^1}$ with $c=1$ based on the generating path from Figure \ref{['fig1']}.
  • Figure 3: Given the linear forest $E_k^{s_k+3}$ (or $F_k^{s_k+3}$, or $G_k^{s_k+3}$) with three edges, there exist at most $5$ edges of type $\phi(s_k+4)$ that can be added to it to produce a graph that is not a linear forest.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2