On \(k\)-step Hamiltonian Graphs

For integers \( k \geq 1 \), a \((p,q)\)-graph \( G = (V, E) \) is said to admit an AL(\(k\))-traversal if there exists a sequence of vertices \((v_1, v_2, \ldots, v_p)\) such that for each \( i = 1, 2, \ldots, p-1 \), the distance between \( v_i \) and \( v_{i+1} \) is \( k \). We call a graph \( k \)-step Hamiltonian (or say it admits a \( k \)-step Hamiltonian tour) if it has an AL(\(k\))-traversal and \( d(v_1, v_p) = k \). In this paper, we investigate the \( k \)-step Hamiltonicity of graphs. In particular, we show that every graph is an induced subgraph of a \( k \)-step Hamiltonian graph for all \( k \geq 2 \).