Hamiltonicity of Token Graphs of some Join Graphs

TL;DR

This paper investigates the Hamiltonian properties of k-token graphs derived from join graphs, revealing an infinite family where these token graphs are Hamiltonian despite the original graphs not being Hamiltonian.

Contribution

It introduces the first known family of non-Hamiltonian graphs whose k-token graphs are Hamiltonian for 2<k<n-2.

Findings

Identifies conditions under which k-token graphs are Hamiltonian

Provides an infinite family of graphs with Hamiltonian token graphs

First example of non-Hamiltonian graphs with Hamiltonian token graphs

Abstract

Let $G$ be a simple graph of order $n$ and let $k$ be an integer such that $1\leq k\leq n-1$. The $k$-token graph $G^{\{k\}}$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $G^{\{k\}}$ whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we study the Hamiltonicity of the $k$-token graphs of some join graphs. As a consequence, we provide an infinite family of graphs (containing Hamiltonian and non-Hamiltonian graphs) for which their $k$-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which their $k$-token graphs are Hamiltonian, for $2<k<n-2$.