Hamiltonian properties of 3-connected {claw,hourglass}-free graphs

We show that some sufficient conditions for hamiltonian properties of claw-free graphs can be substantially strengthened under an additional assumption that G is hourglass-free (where hourglass is the graph with degree sequence 4,2,2,2,2). Let G be a 3-connected claw-free and hourglass-free graph of order n. We show that (i) if G is P₂₀-free, Z₁₈-free, or N_2i,2j,2k-free with i+j+k≤9, then G is hamiltonian,(ii) if G is P₁₂-free, then G is Hamilton-connected,(iii) G contains a cycle of length at least min{σ₁₂(G),n}, unless L⁻¹(cl(G)) has a nontrivial contraction to the Petersen graph,(iv) if σ₁₃(G)≥n+1, then G is hamiltonian, unless L⁻¹(cl(G)) has a nontrivial contraction to the Petersen graph. Here P_i denotes the path on i vertices, Z_i (N_i,j,k) denotes the graph obtained by attaching a path of length i≥1 (three vertex-disjoint paths of lengths i,j,k≥1) to a triangle, σ_k(G) denotes the minimum degree sum over all independent sets of size k, and L⁻¹(cl(G)) is the line graph preimage of the closure of G.