Cyclability, Connectivity and Circumference

In a graph G, subset of vertices $$S \subseteq V(G)$$ is said to be cyclable if there cycle containing the in some order. G k-cyclable any $$k \ge 2$$ cyclable. If k ordered are present common that order, then k-ordered. We show when \le \sqrt{n+3}$$ , graphs also have circumference $$c(G) 2k$$ and this best possible. Furthermore \frac{3n}{4} -1$$ k+2$$ for k-ordered we \min \{n,2k\}$$ . generalize result by Byer et al. [4] on maximum number edges nonhamiltonian k-connected graphs, order $$n 2(k^2+k)$$ with $$|E(G)| > \left( {\begin{array}{c}n-k\\ 2\end{array}}\right) + k^2$$ hamiltonian, moreover extremal unique.