On Hamiltonicity of Claw- And Net-free Graphs

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) Hamiltonian s-path, (2) a Hamiltonian st–path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containing e when G is 2–connected. These results imply that a connected {claw, net}-free graph has a Hamiltonian path and a 2–connected {claw, net}-free graph has a Hamiltonian cycle [3]. Our proofs of (1)−(4) are shorter than the proofs of their corollaries in [3], and provide polynomial-time algorithms for solving the corresponding Hamiltonicity problems.