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 Ryjacek’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: (i) If k=2, δ(H)≥3, and for a given t (1≤t≤4) dt(H)≥tn4, then either H is Hamiltonian or cl(H)=L(G) where G is a graph obtained from K2,3 by replacing each of the degree 2 vertices by a K1,s (s≥1). When t=4 and dt(H)=σ4(H), this proves a conjecture in Frydrych (2001). (ii) If k=3, δ(H)≥24, and for a given t (1≤t≤10) dt(H)>t(n+5)10, then H is Hamiltonian. These bounds on dt(H) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σt and Ut for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max{4p−5,2p+1} are fixed for given p, improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer.