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The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by $$c_\mathrm{ind}(G)$$ . We prove that if G is an r-regular graph of order n, then $$c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$$ and we prove that if G is a cubic, claw-free graph on order n, then $$c_\mathrm{ind}(G) > \frac{13}{20}n$$ and this bound is asymptotically best possible. [ABSTRACT FROM AUTHOR]

If $$G$$ is a claw-free Hamiltonian graph of order $$n$$ and maximum degree $$\Delta $$ with $$\Delta \ge 24$$ , then $$G$$ has cycles of at least $$\min \left\{ n,\left\lceil \frac{3}{2}\Delta \right\rceil \right\} -2$$ many different lengths. [ABSTRACT FROM AUTHOR]