Summary: Let \(k \geq 1\) be an integer and let \(G\) be a graph having a sufficiently large order \(n\). Suppose that \(kn\) is even, the minimum degree of \(G\) is at least \(k+2\), and the degree sum of each pair of nonadjacent vertices in \(G\) is at least \(n+\alpha\), where \(\alpha = 3\) for odd \(k\) and \(\alpha = 4\) for even \(k\). Then \(G\) has a \(k\)-factor (i.e. a \(k\)-regular spanning subgraph) which is edge-disjoint from a given Hamiltonian cycle. The lower bound on the degree condition is sharp. As a consequence, we have an Ore-type condition for graphs to have a \(k\)-factor containing a given Hamiltonian cycle.
For the entire collection see [Zbl 1063.05001].