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

For an integer s1,s2,s3>0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, and Ps3+1 of length s1,s2, and s3, respectively. We determine a family F of graphs such that, every 3-connected (K1,3,Ns1,s2,1)-free graph Γ with s1+s2+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)Every 3-connected (K1,3,Ns1,s2,s3)-free graph with s1+s2+s3≤9 is hamiltonian.(ii)If G is a 3-connected (K1,3,Ns1,s2,0)-free graph with s1+s2≤9, then G is hamiltonian if and only if the closure of G is not the line graph of a member in F.(iii)Every 3-connected (K1,3,Ns1,s2,0)-free graph with s1+s2≤8 is hamiltonian.