k -Factors and Neighbourhoods of Independent Sets in Graphs

Sufficient conditions are proved for the existence of a k-factor or a Hamiltonian circuit in a graph G. The main element of each condition is a statement to the effect that |N(X)| ⩾ a|X| + b|V(G)| + c for every non-empty independent subset X of V(G). The theorems are shown to contain various known results.