Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erdős proved that if a graph G with n vertices satisfies e(G)>maxn−k2+k2,⌈(n+1)∕2⌉2+n−122,where the minimum degree δ(G)≥k and 1≤k≤(n−1)∕2, then it is Hamiltonian. For n≥2k+1, let Enk=Kk∨(kK1+Kn−2k), where “∨” is the “join” operation. One can observe e(Enk)=n−k2+k2 and Enk is not Hamiltonian. As Enk contains induced claws for k≥2, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with the largest possible number of edges. We answer this question completely by proving a claw-free analog of Erdős' theorem. Moreover, as byproducts, we establish several tight spectral conditions for a 2-connected claw-free graph to be Hamiltonian. Similar results for the traceability of connected claw-free graphs are also obtained. Our tools include Ryjáček's claw-free closure theory and Brousek's characterization of minimal 2-connected claw-free non-Hamiltonian graphs.