Hamilton cycles in 3-connected claw-free and net-free graphs

For an integer s 1 , s 2 , s 3 > 0 , let N s 1 , s 2 , s 3 denote the graph obtained by identifying each vertex of a K 3 with an end vertex of three disjoint paths P s 1 + 1 , P s 2 + 1 , and P s 3 + 1 of length s 1 , s 2 , and s 3 , respectively. We determine a family F of graphs such that, every 3-connected ( K 1 , 3 , N s 1 , s 2 , 1 ) -free graph Γ with s 1 + s 2 + 1 ≤ 10 is hamiltonian if and only if the closure of Γ is L ( G ) for some graph G ∉ F . We also obtain the following results. (i) 3-connected ( K 1 , 3 , N s 1 , s 2 , s 3 ) -free graph with s 1 + s 2 + s 3 ≤ 9 is hamiltonian. s a 3-connected ( K 1 , 3 , N s 1 , s 2 , 0 ) -free graph with s 1 + s 2 ≤ 9 , then G is hamiltonian if and only if the closure of G is not the line graph of a member in F . 3-connected ( K 1 , 3 , N s 1 , s 2 , 0 ) -free graph with s 1 + s 2 ≤ 8 is hamiltonian.