Let G be a 2-edge connected graph with a t least 5 vertices. For any given vertices a, b, c, and din G with a # b, there exists in G3 a hamiltonian path with endpoints a and b avoiding the edge cd, and there exists in G3 U {cd} a hamiltonian path with endpoints a and b and containing the edge cd. Also, after removal of two edges or one edge and one vertex from G3, the resulting graph is still hamiltonian.
For every positive integer k , the kth power of a graph C , denoted Gk, is the graph having the same vertex set as G , and in which two vertices are adjacent if and only if they are at distance at most k in G . A hamiltonian cycle (respectively, path) in a graph G is a cycle (respectively, path) covering all the vertices of G and G is said to be hamiltonian (respectively, harniltonian-connected) if it contains a hamiltonian cycle (respectively, hamiltonian paths with any prescribed endpoints).
Sekanina [ 6 ] proved that the cube of any connected graph G with at least three vertices is hamiltonian and hamiltonian-connected (this last result was proved independently by Karaganis [ 2 ] ) . Chartrand and Kapoor [ 11 proved that, for any connected graph of order at least 4 and any vertex w in G , G' -w is hamiltonian. Similarly we proved in [4] that if G is a connected graph of order at least 4 and e is an edge of G 3 , then G 3e is a hamiltonian graph. All these results are best possible in the sense that there exist connected graphs whose cubes are not hamiltonian after removal of two vertices or two edges of the cube, or do not contain a hamiltonian cycle containing two prescribed edges.
However, Schaar proved in [5] that the cube of a 2-edge connected graph of order at least 5 is still harniltonian after removal of two vertices. In this paper, we prove similar results; in particular, we show that the cube of any 2-edge connected graph of order at least 5 is still hamiltonian after removal of one