Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free 2-factor. The toughness condition of result sharp. Moreover, for any $\varepsilon>0$ there exists $(2-\varepsilon)$-tough $2P_5$-free without This implies the $P_4 P_{10}$ best possible forbidden in sense.