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

For an integer s1,s2,s3>0s1,s2,s3>0, let Ns1,s2,s3Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3K3 with an end vertex of three disjoint paths Ps1+1Ps1+1, Ps2+1Ps2+1, and Ps3+1Ps3+1 of length s1,s2s1,s2, and s3s3, respectively. We determine a family FF of graphs such that, every 3-connected (K1,3,Ns1,s2,1)(K1,3,Ns1,s2,1)-free graph ΓΓ with s1+s2+1≤10s1+s2+1≤10 is hamiltonian if and only if the closure of ΓΓ is L(G)L(G) for some graph G∉FG∉F. We also obtain the following results. (i)Every 3-connected (K1,3,Ns1,s2,s3)(K1,3,Ns1,s2,s3)-free graph with s1+s2+s3≤9s1+s2+s3≤9 is hamiltonian.(ii)If GG is a 3-connected (K1,3,Ns1,s2,0)(K1,3,Ns1,s2,0)-free graph with s1+s2≤9s1+s2≤9, then GG is hamiltonian if and only if the closure of GG is not the line graph of a member in FF.(iii)Every 3-connected (K1,3,Ns1,s2,0)(K1,3,Ns1,s2,0)-free graph with s1+s2≤8s1+s2≤8 is hamiltonian.