Hamiltonian Cycles in Tough $(P_2\cup P_3)$-Free Graphs

Let $t>0$ be a real number and $G$ graph. We say is $t$-tough if for every cutset $S$ of $G$, the ratio $|S|$ to components $G-S$ at least $t$. Determining toughness an NP-hard problem arbitrary graphs. The Toughness Conjecture Chv\'atal, stating that there exists constant $t_0$ such $t_0$-tough graph with three vertices hamiltonian, still open in general. A called $(P_2\cup P_3)$-free it does not contain any induced subgraph isomorphic $P_2\cup P_3$, union two vertex-disjoint paths order 2 3, respectively. In this paper, we show 15-tough hamiltonian.