Cyclability, Connectivity and Circumference
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In a graph G, a subset of vertices S ⊆ V(G) is said to be cyclable if there is a cycle containing the vertices in some order. G is said to be k-cyclable if any subset of k ≥ 2 vertices is cyclable. If any k ordered vertices are present in a common cycle in that order, then the graph is said to be k-ordered. We show that when k ≤√(n+3), k-cyclable graphs also have circumference c(G) ≥ 2k, and that this is best possible. Furthermore when k ≤3n/4 -1, c(G) ≥ k+2, and for k-ordered graphs we show c(G) ≥min{n,2k}. We also generalize a result by Byer et al. on the maximum number of edges in nonhamiltonian k-connected graphs, and show that if G is a k-connected graph of order n ≥ 2(k^2+k) with |E(G)| > n-k2 + k^2, then the graph is hamiltonian, and moreover the extremal graphs are unique.
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