On graphs whose square have strong hamiltonian properties

The square  G2G2 of a graph GG is the graph having the same vertex set as GG and two vertices are adjacent if and only if they are at distance at most 2 from each other. It is known that if GG has no cut-vertex, then G2G2 is Hamilton-connected (see [G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor, C.St.J.A. Nash-Williams, The square of a block is hamiltonian connected, J. Combin. Theory Ser. B 16 (1974) 290–292; R.J. Faudree and R.H. Schelp, The square of a block is strongly path connected, J. Combin. Theory Ser. B 20 (1976) 47–61]). We prove that if GG has only one cut-vertex, then G2G2 is Hamilton-connected. In the case that GG has only two cut-vertices, we prove that if the block that contains the two cut-vertices is hamiltonian, then G2G2 is Hamilton-connected. Further, we characterize all graphs with at most one cycle having Hamilton-connected square.