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

Abstract

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 N2i,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 Pi denotes the path on i vertices, Zi (Ni,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, σk(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.