Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ every $(K_1\cup P_2)$-free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; $(\ast2)$ every 1-tough $(K_1\cup P_3)$-free graph is hamiltonian. We conjecure that every 1-tough $(K_1\cup P_4)$-free graph is hamiltonian and every 1-tough $P_4$-free graph is hamiltonian