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 $$C$$C is a hamiltonian cycle of a graph $$G$$G, then the square of $$C$$C is called a hamiltonian square of $$G$$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 $$K_3$$K3 and $$K_4$$K4, the square of a cycle is always 4-connected.

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