Hamiltonicity of Token Graphs of Some Join Graphs

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