Independence numbers of the 2-token graphs of some join graphs

TL;DR

This work studies the independence numbers of 2-token graphs $F_2(G)$ when $G$ is a join graph, particularly focusing on $G=E_n+H$. The authors introduce a general construction that, from an independent set in $F_2(G)$, yields an independent set of size at least as large, enabling exact calculations for several natural join-graph families. They establish a foundational lower bound for join graphs and provide an exact formula for $\,\alpha(F_2(P_m - A))$, which is then leveraged to obtain closed forms for $F_2(F_{n,m})$, $F_2(W_{n,m})$, and $F_2(E_n+K_m)$, with additional results for $F_2(K_{n,m})$. The results give precise independence numbers and illustrate a robust method potentially applicable to broader join-graph classes, contributing to the broader understanding of token graphs and their independence structure.

Abstract

The $2$-token graph $F_2(G)$ of a graph $G$ is the graph whose set of vertices consists of all the $2$-subsets of $V(G)$, where two vertices are adjacent if and only if their symmetric difference is an edge in $G$. Let $G$ be the join graph of $E_n$ and $H$, where $H$ is any graph. In this paper, we give a method to construct an independent set ${\mathcal I}'$ of $F_2(G)$ from an independent set ${\mathcal I}$ of $F_2(G)$ such that $|{\mathcal I}'| \geq |{\mathcal I}|$. As an application, we obtain the independence number of the $2$-token graphs of fan graphs $F_{n, m}$, wheel graphs $W_{n, m}$ and $E_n+K_n$.

Theorems & Definitions (16)

• Theorem 1
• Proposition 2
• Definition 1
• Lemma 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Claim 6