Characterizing Forbidden Pairs for Hamiltonian Squares

The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if C is a hamiltonian cycle of a graph G, then the square of C is called a hamiltonian square of G. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square, in a 4-connected graph. The connectivity condition is necessary as, except K3 and K4, the square of a cycle is always 4-connected.