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Exploring the 3-Token Graph of Particular Graphs

Felicia Servina Djuang, Arizka Yuliana, Widi Bagaskara, Yeni Susanti

TL;DR

This work addresses the structure of $3$-token graphs derived from path graphs and their behavior under disjoint unions. It develops a decomposition framework for $Γ_3(G ⊕ H)$, establishes the isomorphism $Γ_3(P_n) ≅ CS_n$, and analyzes automorphisms and key invariants such as diameter and independence. A general structure theorem for disjoint unions is provided, along with an explicit connection to the cubical staircase $CS_n$ and a complete automorphism description for $Γ_3(P_n)$, plus conjectures for edge-independence. The results deepen token-graph theory and offer tools for exploring higher $k$-token graphs and related graph classes.

Abstract

This study investigates the properties of the 3-token graph derived from path graphs, with a particular focus on its structural characteristics and key attributes. We analyze how the 3-token graph is constructed from path graphs and explore fundamental properties such as connectivity, diameter, and chromatic number. Furthermore, we extend our analysis to the 3-token graph of the disjoint union of two given graphs, examining its unique features and how the structure of the original graphs influences the resulting 3-token graph. The findings of this study contribute to a deeper understanding of token graphs and their applications in graph theory. (We would like to note that an earlier version of this manuscript was previously made available as a preprint on Preprints.org (DOI: 10.20944/preprints202505.1605.v1). The current submission corresponds to the revised version that has been uploaded to arXiv, in accordance with the journal's requirement for preprint deposition. We confirm that both versions refer to the same work and no duplicate submission is intended.)

Exploring the 3-Token Graph of Particular Graphs

TL;DR

This work addresses the structure of -token graphs derived from path graphs and their behavior under disjoint unions. It develops a decomposition framework for , establishes the isomorphism , and analyzes automorphisms and key invariants such as diameter and independence. A general structure theorem for disjoint unions is provided, along with an explicit connection to the cubical staircase and a complete automorphism description for , plus conjectures for edge-independence. The results deepen token-graph theory and offer tools for exploring higher -token graphs and related graph classes.

Abstract

This study investigates the properties of the 3-token graph derived from path graphs, with a particular focus on its structural characteristics and key attributes. We analyze how the 3-token graph is constructed from path graphs and explore fundamental properties such as connectivity, diameter, and chromatic number. Furthermore, we extend our analysis to the 3-token graph of the disjoint union of two given graphs, examining its unique features and how the structure of the original graphs influences the resulting 3-token graph. The findings of this study contribute to a deeper understanding of token graphs and their applications in graph theory. (We would like to note that an earlier version of this manuscript was previously made available as a preprint on Preprints.org (DOI: 10.20944/preprints202505.1605.v1). The current submission corresponds to the revised version that has been uploaded to arXiv, in accordance with the journal's requirement for preprint deposition. We confirm that both versions refer to the same work and no duplicate submission is intended.)

Paper Structure

This paper contains 2 sections, 13 theorems, 49 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Results

Key Result

Lemma 1

Given two path graphs $P_n^1$ and $P_n^2$, with $n \geq 3$. Let $2P_n=P_n^1 \oplus P_n^2$. It follows that

Figures (5)

  • Figure 1: The Graph $\Gamma_3(2P_4)$
  • Figure 2: The Graph $\Gamma_3(P_4 \oplus C_3)$
  • Figure 3: The Graph $CS_8$
  • Figure 4: Graphs $CS_4$, $CS_5$, $CS_6$, and $CS_7$
  • Figure 5: Graph $CS_6$

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Definition 1
  • Lemma 2
  • proof
  • ...and 17 more