Hamiltonicity of Token Graphs of Some Join Graphs

Let G be a simple graph of order n with vertex set V(G) and edge set E(G), and let k be an integer such that 1 <= k <= 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 A and B are adjacent in G{k} whenever their symmetric difference AB, defined as (A \ B) boolean OR (B \ A), is a pair {a,b} of adjacent vertices in G. In this paper we study the Hamiltonicity of the k-token graphs of some join graphs. 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 it is proven the Hamiltonicity of their k-token graphs, for any 2 < k < n-2.