Cycles intersecting a prescribed vertex set

A graph is said to have property P ( k,l )( k ⩾ l ) if for any X ∈ ( G k ) there exists a cycle such that | X ∩ V ( C )| = l. Obviously an n ‐connected graph ( n ⩾ 2) satisfies P ( n,n ). In this paper, we study parameters k and l such that every n ‐connected graph satisfies P ( k,l ). We show that for r = 1 or 2 every n ‐connected graph satisfies P ( n + r,n ). For r = 3, there are infinitely many 3‐connected graphs that do not satisfy P (6,3). However, if n ⩾ max{3,(2 r −1)( r + 1)}, then every n ‐connected graph satisfies P ( n + r,n ).