A \([k,k+1]\)-factor containing a given Hamiltonian cycle (Q5906318)

An \([a,b]\)-factor of a graph \(G\) is a spanning subgraph \(F\) such that the degree of each vertex of \(F\) is in the interval \([a, b]\). For a positive integer \(k \geq 2\), let \(G\) be a graph of order \(n\) with \(n \geq 8k - 16\) if \(n\) is even and \(n \geq 6k - 13\) if \(n\) is odd. If the sum of the degrees of each pair of nonadjacent vertices of \(G\) is at least \(n\), then \(G\) contains a \([k, k+1]\)-factor. Examples are given to show that the results are sharp for both \(n\) odd and \(n\) even.