Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erd\H{o}s proved that if a graph $G$ with $n$ vertices satisfies $$e(G)>\max \{\left(\genfrac{}{}{0px}{}{n-k}{2}\right)+k^2,\left(\genfrac{}{}{0px}{}{\lceil (n+1)\mathrm{/}2\rceil }{2}\right)+{\lfloor \frac{n-1}{2}\rfloor }^2\},$$ where the minimum degree $\delta (G)\ge k$ and $1\le k\le (n-1)\mathrm{/}2$, then it is Hamiltonian. For $n \geq 2k+1$,let$E^k_n=K_{k}\vee (kK_1+K_{n-2k})$,where\mathrm{"}$\vee$" is the “join” operation. One can observe $e(E_n^k)=\left(\genfrac{}{}{0px}{}{n-k}{2}\right)+k^2$ and $E_n^k$ is not Hamiltonian. As $E_n^k$ contains induced claws for $k\ge 2$, 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 Erd\H{o}s’ 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 Ryj'a\v{c}ek’s claw-free closure theory and Brousek’s characterization of minimal 2-connected claw-free non-Hamiltonian graphs.