On a Hamiltonian cycle in which specified vertices are not isolated (Q1850042)

Let \(G\) be a graph of order \(n\) and \(B\) a set of at least \(3n/4\) vertices. It is shown that if \(\delta(G) \geq n/2\), then \(G\) contains a Hamiltonian cycle such that each vertex of \(B\) is adjacent on the Hamiltonian cycle to some other vertex of \(B\). The condition \(\delta(G) \geq n/2\) is needed for the existence of a Hamiltonian cycle and \(|B|\geq 3n/4\) is needed to avoid an isolated vertex of \(B\) on the Hamiltonian cycle.