On 3-step Hamiltonicity of certain graphs

A (p, q) -graph G with vertex set V(G) and edge set E(G) is said to be AL(k) -traversable for k ≥ 1 if we can arrange its vertex set as the sequence of vertices {v₁,v₂, …, v_p} such that the distance between v_i and v_i+1 for each i = 1,2, …, p -1 is k. A graph G is called k -step Hamiltonian if it is AL(k) -traversable and d(v₁,v_p) = k. Then, the sequence v₁,v₂,…,v_p,v₁ is called a k -step Hamiltonian cycle of G. In this paper, we investigate 3 -step Hamiltonicity of certain graphs and its line graphs.