Graphs with every k-matching in a Hamiltonian cycle

Abstract 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 have d(x,y)=2⇒ max {d(x),d(y)}⩾ n+k 2 and there is a k-matching M ⊂ G , ( k ⩾0) which is not contained in any Hamiltonian cycle of G.