Degree and neighborhood conditions for hamiltonicity of claw-free graphs.

For a graph H, let σt(H)=min{Σi=1tdH(vi)|{v1,v2,…,vt}is an independent set in H} and let Ut(H)=min{|⋃i=1tNH(vi)||{v1,v2,⋯,vt}is an independent set in H}. We show that for a given number ϵ and given integers p≥t>0, k∈{2,3} and N=N(p,ϵ), if H is a k-connected claw-free graph of order n>N with δ(H)≥3 and its Ryjác̆ek’s closure cl(H)=L(G), and if dt(H)≥t(n+ϵ)∕p where dt(H)∈{σt(H),Ut(H)}, then either H is Hamiltonian or G, the preimage of L(G), can be contracted to a k-edge-connected K3-free graph of order at most max{4p−5,2p+1} and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: