Graphs with every k-matching in a Hamiltonian cycle Original Research Article

Using the property that being s-edge-Hamiltonian is (n+s)-stable, we characterize all 3-connected graphs G of order n⩾3, such that for all vertices x,y∈V(G) we haved(x,y)=2⇒max{d(x),d(y)}⩾n+k2and there is a k-matching M⊂G, (k⩾0) which is not contained in any Hamiltonian cycle of G.